We paid in to MO state (non-resident), but I'm supposed to mail something to my home state (MS) to receive a credit for that. What do I mail to receive the credit?

Mail a copy of the Missouri return.  What MS wants to verify is that you are filing a MO return that has a tax liability for at least the amount of credit you are claiming in MS.  The MO return (not the MO W2) has that information.