What is a good introduction in gradient flows in metric spaces? I know the book Gradient flows: in metric spaces and in the space of probability measures by Luigi Ambrosio, Nicola Gigli and Giuseppe Savaré, but is too hard for an introduction (for me). I'm looking for something with a similar content.
Here are some links to the online lecture notes which are hopefully more accessible than the book you mentioned:
The book Topics in Optimal Transportation by Cédric Villani is not exactly on this topic but is very well written and contains a lot of related material good for background, motivation and applications. The book of Ambrosio, Gigli and Savaré is indeed pretty dry, but the results they established improved considerably on what was available in the literature.
The notes of Daneri and Savaré look good --- Savaré's presentations in a summer school this past June are available here.