Learn French: Il a expliqué qu'il n'y avait pas eu de dégâts. - He explained that there had been no damage.

This is the sequence of tenses in reported speech. The original direct quote was likely in the passé composé: Il n'y a pas eu de dégâts (There was no damage). Because the reporting verb a expliqué is in the past, the reported verb takes a step further back into the past, shifting from the passé composé to the plus-que-parfait (avait eu).

How does the il y a (there is) structure work in the plus-que-parfait here?

The plus-que-parfait is formed with a helping verb in the imperfect tense plus a past participle. For il y a, the main verb is avoir. So, you take the imperfect of avoir (il y avait) and add the past participle of avoir (eu). This gives you il y avait eu (there had been).

Where exactly do the negative words ne and pas go in this long phrase?

In compound tenses, ne and pas hug the conjugated helping verb. Here, the helping verb is avait, and the pronoun y must stay glued to the front of it. So the negation wraps around that whole block, giving us n'y avait pas, while the past participle eu sits outside the sandwich at the very end.

Why is it de dégâts instead of des dégâts?