"The Feels" is a piece of internet slang of recent coining. As could perhaps be guessed, the term is short for "the feelings", with the additional connotation of feelings that are a mixture of sentimental, heartwarming and mournful. The term "The Feels" is often associated with fandom, tumblr and animated gifs.
However, I have never seen the term "The Feels" used non-ironically, and I shudder to think what type of person would earnestly talk about how an animated gif from Doctor Who "gives them all the feels".