Because here जा means went . टोपी एक डाली पर जा लटकी। it means Hat hanging on a branch. Here जा represent that hat has been hanged as it went their.

Let's say, I am playing with my hat(throwing it up and catching it); suddenly I became unmindful and it went up and stuck in a branch of a tree nearby. In that case, I will say, "टोपी एक डाली पर जा(went) लटकी।". As the hat unintentionally stuck in a branch, I will use टोपी एक डाली पर जा (के) लटकी। I think you got this in a story, right?

Check out my hindi song that i just released yeaterday. watch and comment. Watch An African singing hindi/ Indian song... https://www.youtube.com/watch?v=frFcD9kw1ME

It is used to say that the hat ended up on the branch.