What is the purpose of the two rays -W_i and W_i? Am I correct to assume that the former is reflected light from the object whereas the latter is the source of light?
The transport function tr(p,w) returns a surface point. So Lo(tr(p, wi), -wi) = Lo(p', -wi), which is equal to Li(p, wi), meaning that the "amount of light" concentrating at a particular point p' toward direction -wi is equal to that at point p going at direction wi. I don't think there are explicit "source" and "reflection" here, but just a way of calculating light transfer.
I don't really understand this part... why the reflected light from the object equals to the source of light. Where are those cos theta terms