Electric range stove burned black hot wire
The #10 wire is too small coming from a 50 amp breaker. #10 or 10awg wire is only rated for 30 amps by code - this is why someone advised you to change to #6.

Just to make sure we are talking about the same wire, there should be #6 or #8 coming from the breaker to the range. If the wire that burned off is inside the range going to let's say an element or burner, then stay with the same size wire (high temp fixture wire).  Since you are talking about #14 I am guessing it goes to an element. The important thing about connections at elements is that you use connectors rated for high temp, for example many oven elements are male spade disconnect type connectors. Using a standard female crimp connector will have you back in there because the standard ones can not handle the heat. The ones I use on ovens are rated for 900f. So when replacing the wire if crimp connectors are used at the element use high temp ones or splice a piece on the existing wire using a high temp wire nut.