Textbooks or notes on gradient flows in metric spaces

Here are some links to the online lecture notes which are hopefully more accessible than the book you mentioned:

• Lecture Notes on Gradient Flows and Optimal Transport by S. Daneri

• An Introduction to Gradient Flows in Metric Spaces and Introduction to Gradient Flows in Metric Spaces (II) by P. Clément

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.