Why do electron donating groups stabilize free radicals?
Throughout this answer, I'm assuming that you mean carbon free radicals, as you tagged this with organic chemistry.

A free radical is actually more than just a molecule. It is somewhere between an ion and a stable molecule, but has no formal charge. However, it is highly reactive, as you mentioned. You've also identified the right reason for the need of electrons, it is the octet rule, at least at an elementary level. They behave as electrophiles, in order to complete their octet.

A general free radical of carbon has 7 valence electrons, (like the methyl, ethyl free radical etc.) and while other free radicals of carbon do exist (like carbene, benzyne, and a few I don't know about), most of these are hypo octet, i.e. they have fewer electrons than necessary to complete their octet.

Hence, they get stabilised by electron releasing groups, which "quench" the requirement of electrons for the free radicals. This can happen via induction, or by resonance,or even by hyperconjugation (which is basically the same as resonance).

I hope this answers your question.