A friend of mine mentioned that testing hypothesis in social sciences the "classical" way can be quite limiting and that
Bayesian Statistics are "the future".
Could you explain why that is?
Do you have any recommendations on a good textbook?
E. T. Jaynes' book "Probability Theory" is a tour de force and freely available[1]. The preface and first chapter answers your first question; it exceeds anything you'll probably read here, certainly from me.
Sounds like your friend is an acolyte of the LessWrong cult. That doesn't mean your friend is wrong. If you go on Lesswrong's website they have all sorts of articles on why Bayesian statistics is better.
I haven't looked into it very much, but I think they see Bayesian statistics as better because it better takes prior information into account.
I'm still learning about Bayesian Stats myself, but so far I've really enjoyed Think Bayes[0] by Allen Downey. He's a great writer and one of his other books was recently on the front page here.
[1] http://bayes.wustl.edu/etj/prob/book.pdf