When you put a pot of water on the stove and turn the burner on, the heat source of the stove gets hot relatively fast, but the water takes a while to boil. For multiple reasons it takes a while for the water to heat up*, or in other words for the heat of the burner to be transferred throughout the pot of water.
When you add the ice cube to the boiling water, you very quickly cool off the boiling water which will cause the water to stop boiling. It will then take some time for the heat from the burner to heat the water back up to boiling.
There's a lot of caveats to take into account, for example, if you drop a small ice cube into a massive pot of boiling water, you'll only cool of a small bit of the water near the surface, and maybe only at part of the surface, so you may not even notice the stopped boiling.