The Brewers had 25 players on the active roster when they swept the Rockies in the National League Division Series, 22 of whom appeared in games. Of those 25, only four were drafted by Milwaukee and never left: pitchers Corbin Burnes and Brandon Woodruff, shortstop Orlando Arcia, and outfielder Ryan Braun -- and in Braun's case, it's now been over 13 years since he was taken fifth overall in the 2005 Draft.
So it goes for Milwaukee, which has completed a turnaround from 94 losses in 2015 and 89 more in '16 in stunningly quick fashion, in part because general manager David Stearns (and to some extent his predecessor, Doug Melvin, who held the position from 2002-15) has piled moves upon other moves in order to build a roster that enters the National League Championship Series on an 11-game winning streak.
Of the 30 teams, the Brewers were essentially tied with the A's for the lowest percentage of Wins Above Replacement provided this year by homegrown players -- a grand total of 2 WAR, mostly from Braun. (For context: Michael Trout and Mookie Betts just completed 10-win seasons. All homegrown Brewers combined equaled one-fifth of that.)

Your initial reaction might be to think, "That's bad. That this is an indictment of a farm system that admittedly hasn't had much luck with their top Draft picks." (The only Milwaukee first-rounder drafted since 2009 to make an impact is Mitch Haniger, and he's done his best work for Seattle.) Perhaps so. Then again, it shows that this isn't a situation where a small-market team is only succeeding because poor results have led to high Draft picks. It's that prospects can be just as valuable in terms of who they are traded for as they can be producing for their original team.
The Brewers have built this team via waiver claims, like Jesus Aguilar, Hernan Perez, and Junior Guerra. They've signed Major League free agents, like Lorenzo Cain, Wade Miley, Dan Jennings, and Jhoulys Chacin. They've signed international (re)imports, like Eric Thames. They've made purchases, like Erik Kratz.
Milwaukee has taken risks, too, overstuffing its infield during the season and asking third baseman Travis Shaw to play second base, which he's done adequately despite his total lack of experience there. But this is really a story about the Brewers' trades: The ones you know, and the connections you might not realize.
In one sense, it all started when Cain and Jeremy Jeffress first left town, nearly eight years ago. For example, did you know that Cain and Jeffress indirectly played a role in the arrival of Christian Yelich?
Let's connect the dots.
Dec. 19, 2010: Cain, Jeffress, Alcides Escobar, and Jake Odorizzi traded from the Brewers to the Royals for Zack Greinke and Yuniesky Betancourt
This was the trade that formed the nucleus of the 2014-'15 Kansas City pennant winners (Odorizzi went to Tampa Bay as part of the deal to acquire James Shields and Wade Davis), but it's still being felt in Milwaukee, too, and not just because Cain and Jeffress came back. Greinke gave the Brewers a solid year and a half, but that was just the start of it.
July 27, 2012: Grienke traded from the Brewers to the Angels for Johnny Hellweg, Ariel Pena and Jean Segura
Segura actually didn't do much for Milwaukee, hitting .266/.302/.361 in parts of four seasons. Hellweg and Pena didn't do anything at all, really. Still, Segura would provide value to the Brewers in the end regardless. 
Jan. 30, 2016: Segura and Tyler Wagner traded from the Brewers to the D-backs for Isan Diaz, Chase Anderson, and Aaron Hill.
Though Anderson didn't pitch in the NLDS, he's been a key member of the rotation over the last three years, giving Milwaukee a 3.71 ERA in 86 games. While Hill wasn't with the Brewers for long, the name to know there is Diaz.
Aug. 1, 2016: Jeffress and Jonathan Lucroy traded from the Brewers to the Rangers for Lewis Brinson, Luis Ortiz, and Ryan Cordell
Later that year, Jeffress -- who'd returned as a free agent in 2014 and would return yet again in '17 -- and longtime catcher Lucroy were sent to Texas for a trio of prospects. Lucroy is on his third team since leaving the Brewers and has never hit like he did in Milwaukee. Cordell was traded in 2017 for Anthony Swarzak, who was fantastic down the stretch last year, but now you know where Brinson came from, and now you know where Diaz came from, and that's about to be a big deal when you remember who went into the biggest trade the Brewers have made yet. 
Jan. 25, 2018: Diaz, Brinson, Monte Harrison and Jordan Yamamoto traded from the Brewers to the Marlins for Yelich
Yelich is almost certainly going to win the National League Most Valuable Player Award, and his contract runs through 2022, if the Brewers exercise a team option that will see him make less than $13 million per year in that span. It cost them two of their homegrown prospects and two others acquired via trade, and given the early struggles of Harrison and Brinson, it looks like this could potentially go down as a heist.

Before we move on, remember Ortiz, who also arrived in the Lucroy deal? He came in handy, as well.
July 31, 2018: Ortiz, Jonathan Villar, and Jean Carmona traded from the Brewers to the Orioles for Jonathan Schoop
Schoop hasn't done much for Milwaukee, it should be noted, and he received only two plate appearances in the NLDS, though he was an All-Star for the Orioles in 2017. For now, he's an infield depth piece and potentially a starting option in 2019 and came at the cost of Ortiz, Carmona, and Villar, himself acquired via trade from Houston for Cy Sneed in 2015. 
Speaking of trades with Houston, this 2015 deal, often mistakenly credited to Stearns but actually one of Melvin's final moves, has proven to be the gift that keeps on giving. (Stearns played a role, however; he was the Astros assistant GM at the time.)
July 30, 2015: Brewers traded Carlos Gomez and Mike Fiers to Astros for Josh Hader, Brett Phillips, Domingo Santana, and Adrian Houser
This one happened only a couple of days after the infamously canceled trade that would have sent Gomez to the Mets for Wilmer Flores and Zack Wheeler. While Wheeler blossomed into an ace this year, it's incredible to think how different the Brewers would feel without Hader. (Not to mention that the Mets wouldn't have then traded Michael Fulmer and Luis Cessa for Yoenis Cespedes, and then the Tigers wouldn't have had Cessa to move with Chad Green for Justin Wilson, who then went to Chicago for Jeimer Candelario, and so on.)  
Obviously, Hader has become one of the most feared relievers in baseball, and Santana showed he can be an above-average power hitter, though the arrival of Cain and Yelich bumped him from the lineup in 2018. For the same reason, Phillips got only 122 Milwaukee plate appearances in 2017-'18, until he helped in a different way.
July 27, 2018: Brewers trade Phillips and Jorge Lopez to Royals for Mike Moustakas
Moustakas has done for the Brewers exactly what he's always done, which is to say be a roughly league-average player with a decent glove and good power who is somewhat limited by on-base skills. More importantly, by pushing Shaw to second, he's given the lineup another solid bat, as Shaw took the spot of Villar, who had hit only .261/.315/.377.
So those are the big, inter-connected trades; here are several others that stand alone and have had an impact.

Jan. 19, 2015: Brewers trade Yovani Gallardo to Rangers for Corey Knebel, Luis Sardinas, and Marcos Diplan
Gallardo had been Milwaukee's ace for years, and only five pitchers have ever started more games for the Brewers. At the time of the trade, Knebel was coming off an elbow injury and an unimpressive debut; now, he's spent the last two years in high leverage innings for the Brewers. After briefly struggling in August, he struck out 33 in 16 games in September, allowing zero runs. 
Knebel had originally been a Tigers draftee and had been traded to Texas in July of 2014 for ... Joakim Soria, now his teammate in Milwaukee after Stearns picked him up from the White Sox in July. 
July 31, 2015: Brewers trade Gerardo Parra to Orioles for Zach Davies
This one is both good (Davies has made 80 starts for the Brewers, and Parra was an easily replaceable veteran) and bad (Parra had originally been acquired for Haniger and Anthony Banda, a Melvin trade that doesn't look so good right now).
Stearns was hired that September. In December, he set about making his mark.
Dec. 9, 2015: Brewers trade Adam Lind to Mariners for Freddy Peralta, Carlos Herrera, and Daniel Missaki.
Lind had had a nice season for the Brewers in 2015, but it's not hard to find defensively limited first basemen with power -- they'd find just that with Chris Carter in 2016 -- and the prize of this deal ended up being Peralta, who had a shockingly effective partial rookie season in 2018 and doesn't turn 23 until next June. 
Dec. 10, 2015: Brewers acquire Manny Pina as player to be named from a November trade that sent Francisco Rodriguez to Tigers.
Pina has been a competent hitter as a backup catcher, and he's developed a reputation as a strong catcher thanks to his elite pop time.  
Dec. 17, 2015: Brewers send Jason Rogers to Pirates for Keon Broxton and Trey Supak.
Broxton has had an up-and-down Milwaukee career. He did hit 20 homers and steal 21 bases with very good defense in 2017, but an inflated strikeout rate led to him spending much of 2018 in Triple-A. But he did help finish off the Rockies with a home run off Wade Davis, while Rogers had two hits for Pittsburgh and wasn't in affiliated ball in 2018. Trade won.

Dec. 6, 2016: Brewers trade Tyler Thornburg to Red Sox for Shaw, Mauricio Dubon, and Josh Pennington.
Shaw was a relatively unheralded young player coming off a below-average season in 2016. It's safe to say he's been somewhat more than that for Milwaukee, hitting 63 homers in two seasons and seamlessly moving around the diamond. Thornburg rarely stayed healthy for the Brewers and hasn't for the Red Sox either. Dubon, by the way, currently ranks as Milwaukee's No. 7 prospect per MLB Pipeline.
It hasn't always worked out, of course. Milwaukee might wish it still had Haniger, Khris Davis, Scooter Gennett, Will Smith or Ji-Man Choi, all ex-Brewers who found success elsewhere in 2018. Your batting average on moves is never going to be 1.000. They don't have to be. You just have to hit on more than you miss. So far, so good on that.
Stearns, and Melvin before him, have been aggressive in the right ways. They gambled that Cain would continue to be a star, and he was. They didn't overextend on risky free-agent starters Jacob Arrieta, Alex Cobb, Yu Darvish, or Lance Lynn, despite the injury to Jimmy Nelson. They signed Chacin and Miley, they traded for Giovany Gonzalez, and it's mostly worked out. 
The Brewers are headed to only their second LCS since reaching the World Series in 1982. It's not all about Stearns, obviously. It's just hard to see them getting this far without the aggressive moves he's made.