It's hard to make any sense out of what you have written.
The wires are melting, or they have "come away", or they are burning out, but you don't see it happen?
If they are really melting or burning out, there would be a smell of burning, there would be heat, maybe smoke. You would definitely see it happening.
But this isn't going to happen if you are measuring voltage. It might if you try to measure current, or put the leads in the wrong socket of the meter, but most meters have a fuse that would blow if you do that. Only the cheapest meters might have no fuse.
Are you sure you are not stretching them or breaking them by pulling too hard or something? Or are you soldering and letting them touch the soldering iron?
Nothing really makes any sense about any of this.