No, I think Pasha is closer to getting it right. I’m a native English speaker, and I say and hear „I see“ all the time and it can mean „I understand“ but frequently doesn’t. So, you need to learn the nuances of the word. It’s far more nuanced that a simple „I understand.“ In addition to meaning „I understand“ in just the „I comprehend“ sense it means „Oh wow. I didn’t know that!“ If someone just said his friend won one  million dollars at the lotto you might say, „I see. That’s fantastic. Lucky him.“ You see, it’s not just that banal „I understood your words“ concept. 

Interestig question. I am learning English and when I talk to people who also learn this language and I answer "I see" in meaning as "I understand". They often ask "what you see there?", not in meaning as "what you understad?".  

Maybe I wrong but I think it is more relavant  to " I see your point of view" and is not relevant to " I understand quantum  mechanics".