Listening and Empathy - Why "I understand" is NOT an empathic response

Sometimes, when someone else is telling me about their situation, I cut in with a seemingly empathetic, "I totally understand"
9 times out of 10, what I really feel is that I've got that person completely figured out, or I've lived through a similar experience enough that the story-telling triggers the memory of MY experience. 

Here's the problem:

The person was sharing their story and experience with me and instead of listening and asking questions  that would invite the person to expand and go deeper into their story, my mind wandered to my own..
Even worse is seizing a pause in the conversation to, "tell you about my similar experience" or jumping too quickly to advise, "have you tried...." The first is where I take over the conversation with my own story (NOT listening) then, a "have you tried" which is telling someone what to do (advice disguised as a question)
Taking turns can be a great way to be in conversation; notice who is doing more of the talking?   What more can you learn by asking one of my toolbox favorites, "what else?" and "tell me more (about)"

I've been making an effort to speak less and listen more wherever I am; social, work, even conflict.  I admit that it feels STRANGE not to talk as much, and benefit is that I am learning more about the people in my life.