Let's ignore the social and political aspects of her life for a second.
She is upset at life and the things that she has to do in order to 'make it' in life (in her case...sing to entertain her parent's social gathering). She feels out of place and uncomfortable in her own skin. So, in an attempt to escape from the stress of her life, she falls down a rabbit hole, nearly drowns in her own self-pitying tears, meets a mad hatter, experiments with shrooms, abuses some animals by playing a game of croquet, and rescues a baby that turns out to be a pig (shrooms?)-all while the Queen of the land is wanting her head on a platter.
Um...sounds like life after moving out of your parent's house? yeah. same here.
The thing I really love about Alice is that...there she is...brooding about looking like a freak in front of people...but when she get's away from her uncomfortable comfort zone, and stumbles into Wonderland, she realizes that everyone is a little mad. And some people are totally willing to accept it and maybe even boast about it. And that is what makes the difference.
Experiencing other people's madness made Alice ready to face her own madness...and even accept and learn to enjoy it.
Something I think I've been learning my whole life...Enjoy what you do. Bring your personality into every aspect of your life that you can. Rock your madness.