Why do I see more steam when I take the teakettle off the flame?

The “steam” you see is what steam engine enthusiasts call wet steam, because it contains unevaporated water as a mist within the actual water vapor, which is a colorless, odorless gas.

Dry steam does not contain any unevaporated water.

Steam vapor forms at temperatures above the boiling point, normally 212 degrees Fahrenheit. Visible steam is either water that is not yet hot enough to form this vapor or, in the case of your kettle, water that has cooled slightly into water droplets.

When a whistling teakettle is taken off the heat, the whistle gets weaker, because the pressure of the water vapor drops as the heat drops, although the visible water droplets momentarily increase.