It's kind of a self-fulfilling prophecy. The reason homers are important in the playoffs, or more decisive, is because pitching and defense are that much better compared to the regular season. The best teams are in the playoffs, and the best teams tend to have better pitching and defense, so the baseline is already naturally significantly higher. With better pitching and defense, it's harder to string hits together and make s*** happen on the bases, so out-homering the opponent will set you up better for getting wins.
You also can't be a bad hitting team and make the playoffs, generally speaking. So teams that make it will have more prolific power hitters/hit better for power as a team, and it will be manifested into better winning chances when encountering good pitching + defense in the playoffs, against similarly powerful teams.