I hope this is in time for your test. I will try to explain it a few different ways.

First, there is an attraction between the two atoms, if they are close enough to feel that attraction but it is not being 'satisfied' they are in a high energy state, when they move to the equilibrium bond distance they are moving to a lower energy state. So forming a bond releases energy. If they are already where they 'want' to be it takes work to pull them apart.

To model this, hold a rubber band in both of your hands, this simulates the attraction between two atoms, if you are not stretching the band this is the energy minimum. If you want to break the bond, you have to pull the band tight until it breaks, this requires an input of energy. If you stretch the band until it has some tension it has stored potential energy when you release the tension on the band that energy is released (e.g. if you let it pull your hands together till they clap you can hear the released energy).

Secondly, you can think of it this way, nature always tries to move to a low energy state, if breaking bonds released energy, then that would mean that free atoms are more stable than molecules, that would mean that all bonds in molecules would spontaneously break, and new bonds would not spontaneously form.

Third, a graphical explanation, look at the third figure, the energy well diagram. The Y-axis is potential energy, and the X-axis is distance between the nuclei. What this shows is that the free atoms way out to the left are at a relatively high potential energy, as they move closer together they go down into the energy well and reach a minimum, this is the chemical bond. To get out of that (break the bond) you must add energy to 'climb the hill.'

And finally the most simple explanation, like breaking a rubber band you have to do work to break the bond (+ΔH), and the opposite process bond forming has the opposite sign (-ΔH).

Hope this helps.