In this article on Silent London, director Chad Stahelski says that by including the clip of the silent movie at the start of the film, he is letting the audience know that many of the techniques used in John Wick Chapter 2 were invented by silent movies, and that the film itself references various silent movies throughout.
“We want to let you know we’re having fun and we stole this all from silent movie people,” says director Chad Stahelski.
The article then goes on to talk at length about John Wick Chapter 2's references to silent films and the way it has quite clearly taken inspiration from them. It's an interesting read, if you're so inclined.
And if you were wondering, the clip is from a Buster Keaton movie.