To slither is to move forward by using a twisting motion. I can see the correlation with "not giving up" because snakes essentially go forward by moving back and forth. But it's a very roundabout metaphorical association, and certainly not a meaning or definition.

I think the student is just joking and saying what snakes do, not what karate students do

Interesting statement! Snakes "slither" in that they move back and forth in a twisting motion, so my guess would be the teacher is speaking more about their physical movement. Could he be trying to get them to sharpen their physical skills? I would assume he was motivating them to act with precision, but it could possibly be metaphorical as well. As stated before , it would be a quite unusual motivating statement (and one that lacked a clear meaning). Hope that helps! Hey, if you're interested in having conversational lessons with a TEFL certified English teacher, I'm currently accepting new students! It would be fun to talk about this and anything else you're interested in!! Check out my profile here: https://teach.italki.com/teacher/5129645