Ask them about their thing before sharing yours. Say things like "That sounds like it was difficult, how did you manage?" or "Oh wow I'm so sorry that happened are you alright?" Then when you share your thing, start with "I can relate, I felt the same way when ___ happened", but don't give too many details unless they ask. Make sure what you say really does relate, and try to tie it back to them.
The big exceptions are for major events, both positive and negative. Never try to relate to someone's grief (like with a death or a major illness) with advice or a story about your experience, unless they ask. In the big deal sad situations, just be quiet and let them talk. When you do speak, say that you're so sorry for what they're going through. Be honest that you couldn't imagine how they must be feeling. Like literally say "Oh I'm so sorry, that must be so hard, I can't even imagine what you must be going through." Ask if they're OK, but don't press for details. If you want to help them, tell them they can ask you if that need anything.
It's similar with an accomplishment or a milestone. If someone announces an engagement or a promotion or something, just congratulate them and say you've seen how hard they worked for their goal. Let them know you're happy for them. But don't relate it back to yourself. They want to enjoy the moment, and know that you're excited for their happiness.
People don't necessarily need you to understand them fully. They just need to know that you're interested in their experiences. When you get into close relationships, full understating is more important. But that's also a two way street. If the relationship is at a point where deeper understanding is needed, you might find it useful to tell them about your difficulty when it comes to this stuff. Just be up front, say you don't always know the right thing to say, and it's easier for you to get what others are going thru if you can relate it to your own experience. This is technically true for everyone, it's just that some people are better at communicating their feelings than others.