Let's say that we have a nucleus with 7 units of mass. And let's say that the combined mass of its constituents is 10 units. The 3 units contribute toward the "binding energy." However, we do know that when (for example) hydrogen fuses to create helium, energy is also released along with that reaction. Shouldn't we say that: Energy (Mass) of constituents = Energy (Mass) of resulting nucleus + Binding Energy+ Energy Released (as radiation) instead of: Energy (Mass) of constituents = Energy (Mass) of resulting nucleus + Binding Energy?
The released energy is exactly that binding energy. Binding things together lowers the total energy in the system (neglecting kinetic energy for the moment). The excess energy present in the system before must go somewhere (because of the conservation of energy), so it is released in some way (mostly emitting by photons or as kinetic energy).
So in your example, the system first has an energy equivalent of 10 mass units, and after binding only 7. When the binding reaction occurs, suddenly there is an energy excess equivalent to 3 mass units in the system, so this energy (the binding energy) is released in some way.