It affects flight time the same way that it affects driving time.
Cars measure their speed over the ground based on wheel speed. Airplanes can measure their speed over the ground using things like GPS. Either way, it's referenced to the ground. Technically the ground is moving, but it's not perceptible to us and we still would say zero speed means not moving over the ground.
Airplanes primarily measure their speed through the air (airspeed), and the air tends to move with the ground, so the situation is the same. If the air is not moving the same as the ground, it's called wind, which is the primary influence on flight time. Wind is influenced by the earth's rotation, but the rotation doesn't directly influence flight time.
