As someone on the spectrum the way I've learned to deal with this is basically: A) first reacting empathically ("Oh god that sounds horrible", "Are you alright?" etc.) B) then bringing up relating things, but trying to always turn the topic back in the end so they can continue about their thing ("Yeah I once hurt my ankle, that wasn't fun. Your arm must hurt so much") C) trying to downplay the relating story a bit if possible is usually good, to make sure you're signaling that you're not trying to steal the spotlight ("It was hard enough just hopping around for months, I can only imagine how difficult it's to do stuff with only one arm") D) if nothing else seems to work, people tend to like being asked questions about them and the thing ("What did the doctor say?", "How long do you think it'll take to heal?")