In general, a lot of the speakers are pronouncing "del-lent," as if the first "e" were in a closed syllable, instead of being the long e ("day"). I can't remember if this recording was an example of that. (I hope I wasn't wrong about the ending -ent; and if it sounds correct now, that's all to the good!)
I think this is an English thing:
if the verb were passive, "by" would sound correct: "The forests are destroyed by fire."
But, since we have an active verb with an understood subject ("they"), "with" seems better in English: "They destroy the forests with fire ." (I think you could also say, "They destroy the forests by means of fire .")
Both are English translations of the ablative of means structure, in Latin.