It's a funny use of the verb, because "light" in the sense of flying or travel usually means to settle. However, the text clearly says that the birds began to fly.
There is a phrasal verb "to light out" (usually seen in the past form, "we lit out"), which means to leave quickly. That is probably what the writer wanted us to think. Also, the sense of "not heavy" is probably suggested here.
It seems the reason we have seemingly opposite meanings for the one word is that the meanings originate from different words which used to have separate spellings.