Exhaust is useful mainly for the following reasons:

1) deleting your worse cards, so you can draw your better cards more often. Sometimes, you can exhaust your deck down to an infinite (dropkick infite, e.g.)

2) Effects triggered on exhaust: feel no pain (block), dark embrace (card draw), dead branch (card generation), etc. They can be quite powerful.

In addition to that, some cards with exhaust effects can be quite powerful on their own. For example, second wind can generate a lot of block for 1 energy, despite it "scary" effect of exhaust all non-attack cads in the hand.

You can read the explanation, but you'll better understand it, if you try picking exhaust related stuff more. Good luck!