When applied current flows through an inductor or coil, a magnetic field is set up. A back-emf is then created that opposes this applied current equal to the applied current due to Lenz law.

Then, there is a storage of energy within the magnetic field, or that it has saturated the inductor. How can this current then be released?

Also, the back emf that oppose the current flow due to the increase in current from 0 is due to the changing magnetic flux. But how do we allow the applied current overcome the equalling opposing back emf?

Firstly you need to clearly distinguish EMF from current - a current cannot oppose back EMF, only the
applied supply voltage can do that.

The difference between applied voltage and back-EMF voltage is what is available to push current through
the resistance of the winding.

When the supply is first connected the back-EMF jumps to nearly match it, and the current starts to
ramp up (the rate of change of current = -back-EMF / inductance). As the current ramps up
the back-EMF reduces as more net voltage is needed to push the higher current through R, the windings
resistance.

The magnetic field builds up with the current, and the stored energy is proportional to the square of the
current.