#ΔH^° "(reaction)" = ∑ΔH_f^° "(products)" − ∑ΔH_f^° "(reactants)"#. I'm having such a hard time understanding this equation. What exactly is happening?

This is a statement of Hess's Law using heats of formation.

Here's a standard Hess's Law problem

Calculate #ΔH^° "(reaction)"# for the combustion of ammonia:

NH₃(g) + ⁷/₄O₂(g) → NO₂(g) + ³/₂H₂O(g)

given the information

½N₂(g) + O₂(g) →NO₂(g); #ΔH^°# = +34 kJ
H₂(g) + ½O₂(g) → H₂O(g); #ΔH^°# = -242 kJ
½N₂(g) + ³/₂H₂(g) → NH₃(g); #ΔH^°# = -46 kJ

You would write

½N₂(g) + O₂(g) →NO₂(g); #ΔH^°# = +34 kJ
³/₂H₂(g) + ¾O₂(g) → ³/₂H₂O(g); #ΔH^°# = -363 kJ
NH₃(g) →½N₂(g) + ³/₂H₂(g) →; #ΔH^°# = +46 kJ

NH₃(g) + ⁷/₄O₂(g) → NO₂(g) + ³/₂H₂O(g); #ΔH^° "(reaction)"# = (+34 – 363 + 46) kJ = -283 kJ

Note that the given equations are all for the formation of the reactants from their elements. So the #ΔH^°# values are really #ΔH_f^°# values.

The (+34 – 363) kJ term is the sum of the heats of formation of the products.

(-46 kJ) is the sum of the heats of formation of the reactants, since the heat of formation of oxygen is by definition zero.

We can write the answer as

-283 kJ = (34 – 363) kJ – (-46 + 0) kJ

Heat of reaction = sum of heats of reaction of products - sum of heats of reaction of reactants or

#ΔH^° "(reaction)" = ∑ΔH_f^° "(products)" − ∑ΔH_f^° "(reactants)"#

So #ΔH^° "(reaction)"# = [1(34 kJ) + ³/₂(-242 kJ)] – 1(-46 kJ) = -283 kJ