In the process of nuclear fusion, large amounts of energy, at temperatures of approximately 120 million kelvin, are required to join two nuclei into a single, heavier nucleus. why does the process of fusion require so much energy in order to take place?

The reason for this huge amount of energy is that the nuclear fusion requires two nuclei to come close enough to be within the range of the strong nuclear interaction, which is responsible for the fusion. But in order to come so close, the two nuclei have to overcome the electrostatic repulsion between them, which becomes stronger as they get closer. In fact, considering two nuclei of hydrogen (two protons), the barrier that  they have to overcome to reach a distance r is equal to the electric potential energy:

And since the range of the nuclear strong interaction is very short, r must be very small, and so the amount of energy required U can be huge.