The period of Renaissance in Europe kicked off after the Black Plague started disappearing. How did the the plague help in igniting the fire of rebirth?
The question seems to assume a few things: first, a certain time frame, and second, causation. I don't think these assumptions are quite accurate.
First, early Renaissance started before the Black Plague. For example, Dante Alighieri wrote decades before the plague. There were some advances in architecture even before that, circa 12th century. Given that technological and artistic advances often seem to snowball, one should look at the timing of the first changes that started the movement rather than the timing when the highest momentum was achieved.
Second, even though Black Plague influenced Renaissance, it wasn't the cause that started the snowball rolling. There were several important events that planted the seeds before that: bits and pieces of knowledge acquired by osmosis from Arabs during the Crusades in 12th century and plunder of Constantinople in the 13th century that brought into the West pieces of art and wealth and a few surviving scholars.
Black Plague obviously had tremendous influence on the development of Renaissance, even though it didn't trigger it. Too many influences to discuss on this site. I think the biggest influences were the realization that the Church doesn't have all the answers and the renewed appreciation of preciousness human life. There were many more of course, but these seem to be most important for setting the right mind frame, IMHO.