The handle on my barr framing chisel popped off the other day. Having never installed a handle, I'd like some advice on the best way to do it without damaging anything.

If it just "popped off" then re-insert it into the cone.
Most are just press fit in, that is when you pound on your handle with a mallet, you're pressing it in.
When a chisel sits for a while without being used the wood can dry out with the winter dryness and then then handle may shrink up a little.
This is why it is best to always pick up a chisel by the metal cone, until you're sure the handle is well seated.

If it broke that's another thing.

Hello,

A literal interpretation of your description would suggest a combination of factors. One, that for some reason - it might be good to track down what that reason is - the handle has indeed dried and shrunk. And, it can therefore slide further into the socket to the point that it has made contact with the bottom causing it to pop out. So before reinserting it, the small diameter end must also be cut back to allow it to again wedge itself properly into the socket.

Greetings,

Don Wagstaff