why do exothermic reactions have a negative enthalpy change?

ΔH = H(products) - H(reactants)

Enthalpy (H) of a substance is the energy content of the substance. When the total enthalpy (H) of the products is lower than that of the reactants, the excess enthalpy is released from the reaction. In other words, the reaction is exothermic.

As H(products) < H(reactants), ΔH = H(products) - H(reactants) < 0 (i.e. negative)

because we chose to assign the sign of the change that way. positive means incoming, negative means outgoing. The sign is an arbitrary convention. exo=outgoing; negative=outgoing.

Historically, the idea is that if you had to heat the system, you were seen to be adding heat, so the enthalpy change was positive. The point of view that was taken was that of the system: the system gains heat so enthalpy must be positive. This obliges the reverse (loss of heat) to be negative.

The final state has a lower energy than the initial state