In double insured, can someone explain how this works like I’m 5 please?

I’m 23 and still on my mom’s insurance, but I make so little and have a lot of medical expenses and medication to pay for.

For the life of me I can’t figure out how this works!

I know they take the private insurance first. Do I also give my doctors my new insurance? Or is it like a claim file thing once I get my bill?