Why is this happening? I can see some other characters (for example, Faendal) doing this, but seing as Aela is a werewolf herself, why in the world would she get pissy when I turn into a werewolf? What's causing this?

What's causing it? Designer logic~

yup... designer flaw... npcs will auto hostile if you transform in front of them. Only if they are a follower do they remain calm I believe. Which is silly how the companions act if you turn into a werewolf in Jorrvaskr over dinner.

It's that skyrim logical logic. Like how the vampire's at the castle (The volkihar clan) will attack the player character if he/she turns into a vampire lord in front of them

That one got me the most... I have never facepalmed so hard in my life.

Actually, the Companions have their own crime faction which ignores werewolf transformation crimes. However, when you encounter them in the wilderness, their crime faction is changed to the current hold's, which don't ignore transformations, and it's not always properly switched back.

The Volkihar also ignore vampire lord transformation; though there are shaky reports of them attacking, I've never seen it.

Actually, I'd say that's probably one of the worst possible places to do it, given the whelps at least in theory don't know the Big Secret - you find out during your Trial, and even that is not supposed to happen, but Farkas' hand is rather forced by the Silver Hand turning up. Ria, Njada and Athis don't seem to know yet (although I'm guessing the cleaning lady does) so if you turn in front of them, that would seem to be a pretty massive no-no (at least until after you become Harbinger.)

I should have added that Aela is my wife and she was my follower when I turned and she attacked.

followers should ignore your transformations. you didn't happen to strike her did you?

Nope. Just transformed into a werewolf, got a message saying "Your follower has left your service" and she's hostile until she loses me and I turn back to a Breton.

interesting... then again I use UFO follower mod. My memory of not using follower mods doesn't go back that far. I do recall playing on the PS3 and having Lydia as a follower. I transformed which she was fine with but when I roared, she got frightened and I had a red dot on my compass but she just went home. Was sitting calming in breezehome watching my empty bed upon my return "Yes my thane?"

This is happen because she is in guard faction, the same faction with other city guards.

This is make her turn hostile if you commit crime such as turn in to werewolf.