I can't say that I would have known what this expression meant if I hadn't been looking directly at the person using it.
The complete phrase is "conocer algo/alguien como la palma de la mano" . It means to know something, or someone, like the back of one's hand.
The only reason I was able to figure it out (and consequently act like I knew more Spanish than I did) is because they were talking about the city they lived in and pointing to the palm of their hand -- "la palma de la mano".
Here are some examples:
Conozco Miami como la palma de la mano
I know Miami like the back of my hand
Lo conozco como la palma de la mano
I know him like the back of my hand
This phrase works exactly the same in English as it does in Spanish, so you can start using this right away.
¡Espero que te sirva¡