The relationship ΔG <= 0 (i.e. that reactions seek to minimize their free energy) is derived from the second law of thermodynamics (specifically for the case of a reaction occurring at constant pressure). The second law states that the entropy of the universe is always increasing. There are two components to the entropy of the universe which we must consider, the entropy of the system and the entropy of the surroundings. By splitting the entropy of the universe into these two components, we can write the second law as follows:

ΔS_{uni} = ΔS_{sys} + ΔS_{surr} >= 0

From this equation, we can see that there will be trade offs between the entropy of the system and the entropy of the surroundings. In order for the system to lose entropy and become more ordered, the surroundings must gain entropy to compensate and vice versa.

The formula for the Gibbs free energy reflects these trade offs between the entropy of the system and the entropy of the surroundings. Recall that for a reaction occurring at constant pressure, the change in enthalpy is equal to the amount of heat released/absorbed by the system, ΔH = q. Further, remember that the change in enthalpy of the surroundings is given by the following formula: ΔS_{surr} = q/T_{surr}. Therefore, the quantity ΔH/T_{surr} will tell you about the change in entropy of the surroundings.

Given that ΔS_{surr} = -ΔH/T_{surr}, we can plug this value into our expression for the entropy of the universe to give:

ΔS_{uni} = ΔS_{sys} + ΔS_{surr} = ΔS_{sys} - ΔH/T_{surr}

Applying the second law, we obtain the inequality:

ΔS - ΔH/T >= 0

Or, equivalently:

ΔH - TΔS <= 0

Which is exactly the equation that explains why spontaneous reactions tend to minimize their free energy.

From these derivations we can see that exothermic reactions are favorable because they transfer heat to the surroundings and therefore increase the entropy of the surroundings. As you correctly mentioned, you can think of this heat transfer as increasing entropy because you "delocalize" the energy trapped within the system to occupy the much larger surroundings.