An independent clause can stand by itself as a sentence.
A sentence has, at its most basic, a subject and a verb.
Mary is the subject, and she’s the one doing the action – running.
Independent clauses can be a lot more complicated and longer, but that’s the basic idea. Here’s a longer version of the above example:
A friend of mine from grade school who knows no shame, Mary ran from her boyfriend when he found out she’d been hanging out with Tommie.
Mary is still the subject, and she’s still running. The opening clause (“A friend of mine…”) is dependent because it makes no sense on its own. Try reading this out loud:
A friend of mine from grade school who knows no shame.
If someone said that to you, you would think they became distracted at the end and forgot to get to the point. “Yeah, what about your friend?” you might say.