Why does a current carrying conductor keep a magnetic field experience force?

Hint: When a charged particle placed in a magnetic field is in motion, the magnetic field exerts a force on the charged particle called magnetic force. For a current in a conductor, there must be a flow of electrons through the conductor. With these two points, you can explain the given statement.

Complete step by step answer:
We know that an external electric field exerts a force on a charged particle, when the charged particle is placed in the external electric field. Similarly, when a moving charged particle enters a magnetic field, the magnetic field exerts a force on the charged particle called magnetic force.
Note that in case of an electric field it does matter whether the charged particle is moving or is at rest. The electric force will be exerted irrespective of the motion of the particle. However, in case of the magnetic field, the charged particle must be in motion. If the charged particle is at rest, the magnetic field will not exert a force on the particle.
Now, we will be able to understand why a magnetic field exerts a force on a current carrying conductor. We know that current is defined as the rate of flow of charge with respect to time. Therefore, for a current there must be a flow of charges. In case of a conductor, the charges are electrons.
Hence, when there is a current in a conductor the charges (electrons) are in motion. And we know that when a charge moves in a magnetic field, the charge experiences a magnetic force. Hence, the magnetic field will exert force on all the moving electrons, which produces a net force on the conductor.
Hence, a current carrying conductor kept in a magnetic field experience force.

Note:
We can explain the given statement with an alternative concept. For this, we must know that a current carrying conductor acts as a magnet. When a magnet is placed in a magnetic field, the magnet experiences a magnetic force. Therefore, a current carrying conductor will also experience a magnetic force, since it acts as a magnet.