Why there is an induce current in a coil when a coil is compressed in a magnetic field?

compressing a coil in a magnetic field means that there is a relative movement of the the said coil existing in a magnetic field, hence current will be induced. induction law.
Another Answer

'Current' is not induced into a coil. It's voltage that's induced; if the coil forms a closed loop, then current will flow; if there is no closed loop, then no current will flow.

The induced voltage results from either a change in current through the coil, or from the relative movement between the coil and an external magnetic field.