How do enzymes not change the overall energy change of the reaction they're catalysing if they lower the activation energy?
You're on a valley on one side of a mountain (representing some fairly stable state of your reactants), and you'd like to go to a town on the other side at a lower elevation (representing another stable state for your products, at lower energy than your current state: this is an energetically favorable reaction).
You can go over the mountain: high activation energy to climb up to the top, then you go down the slope to eventually get down to the town.
You can go through a pass between mountains instead: much lower activation energy to get over the pass, then you eventually get down to the town.
Whichever path you choose, you're at the same elevation (energy, $\Delta H$ compared to where you started), with the same products (your location in space), but if you took the pass you've gotten there via a different chemical route that was easier to achieve energetically at the peak.
Enzymes let you take the path through the pass rather than over the top of the mountain. They don't change where you started from or where you're going, so they don't change the overall energy of the reaction.