The front wheel rotates without slipping and so the cycle/bike rolls smoothly. The force exerted on the bike in the horizontal direction is converted into the rotational energy. The center of mass of the bike has a momentum same as the bike.
The center of mass of bike is above the ground at a height. The various parts of the bike have translational and rotational kinetic energy. There is a non zero moment about the the point of contact of the tyre with the road.
When the sudden brakes are applied very hard on the front wheel, the wheel stops rotating and literally halts. Let us say it does not slip. There is friction on the road. In this case the center of mass of the bike has still the kinetic energy and velocity. So about the point of contact of bike with the road, there is a non zero moment.
That is, the center of mass of bike is a non zero angular momentum around the bottom most point of wheel. Hence, the bike rotates about that point. It lifts into the air, in a circular way, due to the torque with the
angular velocity = linear velocity / height of center of mass.
That is why the fast speeding bikes do Somarsaults when sudden brakes are applied hard.
This is due to conservation of angular momentum and energy.