Is this a problem which can be fixed at home or is it time for a garage?
Depends on your skill level, facilities and tools. The basic choice is between soldering or crimping. You can also buy a repair kit - just do an ebay search.
However it's done, the key to a lasting repair is to use extra flexible wire for the replacement - the kind used for making test leads is ideal. Replacements made using standard auto cable rarely last more than a year or two.
Soldering provides a better electrical connection than crimping, but soldered joints are brittle and the wire can subsequently break at the end of the soldered section. If you want the best of both worlds, the gold standard is to solder in the patch wires, and then slide a preplaced hard shell connector over the soldered joints, held in place and waterproofed with a silicone sealant - but you'll need to strip back a fair way to gain enough access to do this, and stagger the patches. Done this way, the repair should outlast the car.
If you're not going to do this yourself, then I'd suggest finding an auto electrician rather than a garage - many garages will subcontract such work anyway, so better to cut out the middleman and use a tradesperson with appropriate skills.
This is a long running design weakness in the 500 which should, IMO, have been found during testing and fixed before production started. Fiat have been abysmal in acknowledging the problem and supporting those affected by it. The problem is that the wiring is bent around a sharp angle whenever the hatch is opened or closed - redesign would require the hatch to be retooled (expensive). Using a more flexible cable would have been a reasonable workaround, but Fiat did neither.
