A spinning wheel A is brought into contact with another wheel B...

When wheel $A$ is brought into contact with wheel $B$, kinetic friction acts on wheel $B$ at the point of contact.

This frictional force provides a torque that causes wheel $B$ to start rotating. As wheel $B$ accelerates from rest, its rotational kinetic energy increases.

Mechanical energy is the sum of kinetic energy and potential energy. Since the kinetic energy of wheel $B$ increases while its potential energy remains constant (assuming its center of mass does not change height), the total mechanical energy of wheel $B$ increases.

Furthermore, the relative slipping between the two wheels at the contact point generates heat due to friction. This heat raises the temperature of wheel $B$, which means its internal energy also increases.

Therefore, the internal energy, mechanical energy, and kinetic energy of wheel $B$ all increase.