What is the purpose of this wiring connector?

Miroslav Djuric, I have seen these being used extensively on European equipment. Possible a TUV requirement for approval. Those are simple electrical wire screw terminal connector, but allow a variety of cable sizes and higher amperage to be connected. Yes, you can connect your wires by other means, i.e. soldering. Just keep the amperage in mind and stay away from automotive connectors like spade connectors etc. I do wonder why your washer pulled so many amps to actually overheat your connector:-) Hope this helps, good luck