Doesn't Bitcoin need mining to handle transactions?

Yes, is does. However, the amount of electricity used isn't directly correlated with the number of transactions on the network. It instead follows the number of miners validating those transactions. A empty block validating no transaction takes more or less the same amount of work as a full block to mine. That was GP's point, I think.

That being said, having a lot of transactions going around is probably correlated with Bitcoin getting more popular, which is again probably correlated with more miners coming in, which would indeed lead to higher energy consumption. That's an indirect effect, however.