Astros starters simply did not pitch well or enough for Houston to win a long series
The playoffs are a different animal altogether from the regular season. While teams are fairly willing to let starters try to work through jams in the regular season, there is often a very quick hook in postseason games given the importance of each and every game. There is merit to the practice, but starters still need to do their jobs, throw meaningful innings and give their teams a chance and the Astros' rotation simply fell short of that in the ALCS.
Justin Verlander was completely fine here as he threw at least 5.2 innings in both of his starts in the ALCS and averaged three earned runs in each start. Certainly not dominant, but completely reasonable overall. However, Framber Valdez, Cristian Javier, and Jose Urquidy all had at least one start in the series where they failed to make it out of the third inning with Javier's implosion in Game Seven being particularly noteworthy.
The results themselves were bad enough as you never want to be in a hole as you are pulling your starter from the game, but the rotations' struggles also put a huge strain on the Astros' bullpen over the course of a seven game series. Astros relievers had to throw a LOT of innings against the Rangers and by the end of the series, some guys like Bryan Abreu, Hector Neris, and others very clearly were not as crisp as they were earlier on in the postseason. Plus, dipping into the bullpen when they, too, are struggling in the postseason (more on that in a bit) isn't exactly a recipe for success.