I really don't know the first thing about you.
It is rather easy to understand idiom. When you get to know somebody you start to discover facts (things) about that person. One thing may be what that person likes, another thing may be that person's character. So not knowing the first thing about somebody means that you did not discover even one thing/fact about that person, meaning you know nothing about him or her and that person is alien to you.
Yes it means you don't know anything about the person you're speaking to. Just like I didn't know the first thing about you until I clicked on this forum post.
Hope this helps
I often come across "I don't know a thing about you", but here is the definite article, so I think it's either backed by some context, or it's emphasising that he/she hasn't even started to introduce herself/himself.