In field controlled dc motor, why does the armature current remain unchanged even if it is affected by field flux?
Does the speed of rotor changes to bring the armature current back to
the constant value?? What causes the armature current to not change
from its constant value?

If you have an external field winding that controls the magnetic field of a DC motor, then a higher magnetic field means that the armature's back emf equalizes when the speed is lower. This is the criteria for stabilization of speed.
If the back-emf is too low then the motor armature speeds up until the constant applied armature voltage and the back-emf are at the right level to permit the right amount of armature current needed to generate mechanical power to the load and against friction.
It's all embedded in Faraday's induction equation: \$\text{emf} = -N\dfrac{d\Phi}{dt}\$. In other words, to make emf the right value to drive load and friction, the motor can run slower when the flux is higher because, what is lost in the rate of change of flux is regained by the flux actually being higher.