You get a second character.
Honestly, I wanted to just stop there—that’s broken, full-stop, right there. Stack Exchange is making me write more, though.
Cohorts are nearly as strong as PCs and you get all of that power for the cost of one measly feat. Feats are powerful and valuable, but “having another character” gets you a minimum of three, so you’re net +2 on feats, and that’s ignoring literally everything else about the character. Sure, the cohort is squishier and splitting feats between two characters limits synergy, so it’s not literally the same as +2 feats, but you get so much else on top of that. What are the most valuable things in the game? Feats? You get more of those. Spells known? You can get more of those. Spell slots? More of that. Actions in combat? You get more of those.
Leadership is so powerful that it is probably the strongest choice for 6th-level feat, even for druids, who would otherwise consider Natural Spell a foregone conclusion at that level.
Also, it’s a headache to adjudicate. Technically, the feat requires a whole lot of DM input—which is a big ask since DMs are already, ya know, pretty busy. And since the DM builds the cohort—the PC only gets to roughly describe desired attributes like race and class—there is a ton of DM judgment on the line here. Exceptionally poor choices on the DM’s part can take the feat’s power down massively, but that’s fraught with problems too. And if you go hard enough, you could turn the cohort into a liability, which is just miserable.
Finally, it’s just... kind of nonsense within the framework of the game. The cohort isn’t a minion like an animal companion, familiar, or special mount, that shares a magical bond with and is explicitly subservient to the PC. The cohort’s supposed to a wholly independent character. As such, you shouldn’t be able to get one just by spending a feat—you should have to roleplay that, and it should be handled accordingly, with the character making truly independent choices and quite possibly not accompanying the PC on some of their activities. While making it cost a feat seems like it would balance things out, instead it means that it’s something the PC has “paid for” and is therefore owed. That’s a really problematic dynamic.