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.

Interesting. The PVC (Polyvinyl Chloride) composing the alligator clip insulator is only part PVC chemically, a good percentage of it is plasticizer. That's what you smell from something like a swimming pool liner or beach ball, not the plastic but the plasticizer. PVC can be a majority plasticizer by weight if it has to be very pliable.

It's likely that whatever chemical they used in the insulator plasticizer does not get along with the plastic in the breadboard and has dissolved it a bit. The heat will have contributed to the issue and how quickly it occurred.

Personally, I would not worry too much about it being toxic or anything- however the EU has been paying a lot of attention to regulating plasticizers and other chemically active additives such as fire retardants.

ABS (often used on cheap breadboards) is not resistant to phthalates- the traditional PVC plasticizer- according to this chart. From my above link:

In Western Europe about one million tonnes of phthalates are produced each year, of which approximately 900,000 tonnes are used to plasticize PVC.

The most commonly used plasticizers are phthalates and they are probably not good for you, and especially children

Several phthalates are "plausibly" endocrine disruptors.[24] The long-term health effects of exposure to endocrine disruptors, such as phthalates, are unclear.[25]

Authors of a 2006 study of boys with undescended testis hypothesized that exposure to a combination of phthalates and anti-androgenic pesticides may have contributed to that condition.[26]

A scientific review in 2013 came to the conclusion that epidemiological and in vitro studies generally converge sufficiently to conclude that phthalate anti-androgenicity is plausible in adult men