"Why do monsters have feats?"
As far as I know it's because the designers felt the need to have both characters and monsters purportedly play by the same rules. Why did they feel that this was necessary? I have no idea, as from past experience I've discovered that its both it's bad design philosophy and a waste of time that adds nothing to the monster that you couldn't have added yourself and saved the time of rummaging through 90+ books on the off chance you'd stumble upon, I dunno, something? Check out the orc on page 203 of Monster Manual: his feat is Alertness. Whoop-de-flying-shit, he's got +2 to Listen and Spot. I mean, just...wow. Personally I'd rather have given him Exotic Weapon Proficiency (orc double-axe) or even Weapon Focus. Something remotely useful, ya know? With Alertness he's got--after modifiers--a whole +1 to both skills (meaning that he'd be really handy at detecting a fighter in full plate smashing two helmets together).
This question really just scratches the surface. My question is why build them like characters at all? Why do you as a DM need to go through all the motions to essentially make a fully statted player character even though you only need to know how much damage it can take and how much it can deal? Actually, making a monster is an even bigger pain in the ass because of all the fiddly mechanics that go into it. According to the section in Monster Manual on creating monsters, we start with the type, of which there are 15 options (animal, construct, fey, giant, vermin, etc) that determine Hit Dice, attack progression, base saves, proficiencies, and skill points (you know, like a character class). Types also sometimes give you other set benefits, like low-light vision, immunities, etc. This isn't counting one of over 20 subtypes that add more shit to the pile, as well.
The first problem is what you do if your creature seems to fall into multiple categories, such as a Large humanoid with a strong connection to nature, a "fey-giant" if you will. You can't have both the fey and giant types even though they might make sense, so...what? Pick the "best" one? Argue in the forums about it? Both are for humanoid creatures, and apparently giants don't even need to be Large or larger. Under fey it states "if you want to create a tough fey, you will need to assign formidable physical ability scores and special attacks or defenses to make the creature survivable." This is another problem because special attacks are determined by a combination of ability scores and Hit Dice, so if you give a creature a special attack and have to compensate for a shitty type you might end up making their attack virtually impossible to defend against. There's also the issue that not all types are equal. Dragons and outsiders, for example, get a hefty Hit Die, great attack bonus, good saves all around, lot's of skill points, darkvision, some immunities, and maybe some other things that I'm overlooking. Conversely, humanoid are basically clerics without the class features and proficiencies.
Once you get all the, ahem, basics out of the way you still have to build the rest of the monster by determining Hit Dice (which affects lot's of things), Speed, Armor Class, ability scores (also affects a lot of things), size (which affects some things), attacks, special attacks (which have effects based on a combination of ability scores, Hit Dice, and whatever modifiers you care to pile onto it), special defenses, spend skill points, and finally still choose feats (which might force you to go back and correct things). Some of the mechanics have tables that provide recommended values based on the monster's ability scores and size, which are usually somewhat refined by pegging it's Challenge Rating (which is all too often incorrect) and just setting the numbers so that it hopefully challenges the party without killing them. This confuses me because the whole process acts like I'm legitimately building a character up until you get about halfway in, at which point you just do whatever it takes to get the monster to do whatever it is you want.