What kind of rubber "melts" plastic?
"Melts" isn't the correct term here -- that would involve heat.

The vinyl boot on the alligator clip contains a solvent called a plasticizer that helps keep it pliable. However, it is volatile and its outgassing has also affected the plastic used in the breadboards.

The solvent is a volatile organic compound, and is probably dangerous if concentrated sufficiently, but at the low rate that it outgasses in a reasonably ventilated space, it is generally considered safe.