J115 is the input to the driver board for the general illumination. Power comes from the transformer. The Triacs with large blue/purple heat sinks just to the right and above of J115 control the GI power exiting the driver board on J120, J121 and J119. These connectors are burned on may games as well. It looks like J120 has been repaired/replaced on your game.

They could be getting hot due to a short but that would likely result in a blown fuse. The more likely cause is weak connection between the connector and the pin on the board or the wire and the crimp connection causing high resistance which causes heat.

You need to replace the connector and header pins on the driver board for J115. You might need to add a little extra wire to the burned wires on pins 10, 11 and 12.