D e m o n : Who are you?
B u f f y : I’m Buffy, the Vampire Slayer. And you are?
A N N E, 3x01
Overall, I think Anne is a highly underrated episodes. It's not the best pace-wise, there are no really memorable baddies.
But, this is a really interesting choice in season premiere, because like the last one (When She Was Bad), it really piggybacks on the last season's finale in a way that not all of them do. (The Freshman, for instance.) Anne works very well because we leave the finale of season two with Buffy in conflict with herself. Instead of resolving all the conflicts of the season, we're left with this one unresolved.
Buffy hates herself a little bit. No, not a little bit. She looks in the mirror, she sees Buffy, and she sees Angel's killer. She sees the person who killed the man that she loved more than anything in this world. And she can't stand that. She can't keep looking at that, not even for one second. So, she puts that girl on a shelf.
This is why she can't continue staying in Sunnydale -- because she can't be Buffy. Sunnydale is Buffy. She needs to disappear and be someone else so that she can hide from Buffy and hide from the fact that she hates herself. Completely cleave out that part of herself so that she's someone else entirely and doesn't feel that overwhelming, intolerable guilt over Angel's death. Because, for her, it's not as simple as "it was him or the world." It ends at "I killed him."
By making herself into someone else, she can disassociate from that guilt. She can say no, that was Buffy. I'm Anne. I'm not her at all. I didn't do that. She can even say that she never loved him, because it was Buffy who loved him. She can take all of her pain and put it away on someone else.
That must be nice. So nice, in fact, that it would take something so extreme to get her to choose anything but that. Because, Buffy has escaped her pain, she's left behind all the heartache and guilt and self-loathing by disassociating herself from it. Why would she choose to go back to it? But, that's what she has to do, and that's the problem that Joss had to tackle in writing Anne.
And it all comes down to choice, which is a big overarching theme for Buffy this season. She doesn't like other people making choices for her -- we have bookends this season. Very subtle but very present bookends. She starts the season with Anne, rejecting other people taking her identity from her, despite the fact that it was an identity she sacrificed, and she ends the season with Graduation Day. Graduation from the Council, who has been taking everything that she is and forcing themselves on her from day one.
Buffy has chosen to cast aside her identity at this point, but what's important is that it was her who chose to do so. Buffy, whether it's overt or not, has a lot of issue with authority figures. She doesn't like cops, she hasn't liked any of her principals, she patronizes her mother, Giles has had to fight for her respect tooth and nail before finally earning it through immense trial. She doesn't just accept it, because she needs to be the authority -- this is what makes her the Slayer that she is. This is what makes her capable of being a general instead of just a Slayer.
The obvious answer to get Buffy to willingly take back all that pain and heartache was to show her what it felt like to have it taken from her against her will. So, she was faced with demons who wanted to take her identity. Who wanted to brainwash her into believing that she's no one. And this is what it took -- and it's not just a matter of contrariness, although there's a definite factor. It's a matter of it had to be this extreme example of how it is not always sunshine and daisies to lose your identity. Because, you aren't just losing the bad stuff. You're losing yourself entirely. You're a mindless drone, a zombie (although those won't crop up until Dead Man's Party).
So, these demons and their sweatshops in an alternate dimension (nice subtle commentary, Joss) act as a way to jerk Buffy back awake and help her realize that no. She doesn't want to lose Buffy. That the pain and the heartache and the guilt and the self-loathing are parts of being someone. And, she wants to be someone. Being someone matters, and that's the one thing that no one can take away from her.
Another interesting tidbit: up until this point we've constantly seen Buffy in a battle with herself over wanting a normal life versus being the Slayer, and that all comes to a head this season, and I think Anne can be considered the turning point. When she accepted Buffy back, when she took that name back, she accepted that she wanted everything to go with it. She was done behaving as though the Slayer was something she didn't want to be. She wants the whole Buffy package, and her Slayer duties are part of it.
This is another conflict that hung over from the last season (her speech to her mother in Becoming Part II pretty much exemplifies this, because it's the note she leaves her on: "I don't want to be the Slayer, but I have to") that gets tidied up, just in time for Faith to come in and threaten it. Now that Buffy is secure and realizes she wants her life, she has Faith coming in and threatening it by horning in on that part of it. That's meta for another time, but it's very clear that Faith's timing is no coincidence.
Buffy overcomes her rejection of her identity just in time to have someone else trying to take it from her? The crises will never end!
Anyway. TL;DR version, I think Anne is perfect at what it does, and I think that what it does is very underrated. Because it really thrusts Buffy into a wake-up call about the fact that she can't disassociate her pain and guilt about Angel to escape it. She can't keep running and hiding from her identity as the Slayer, either. She has to not only accept both, but seek them out willingly, and fight for them.