Paint it as a chemical reaction in order to understand its equilibrium state. We basically have:

H~2~O (gas) ⇌ H~2~O (liquid)

By sealing the jar with the water already boiling, we initialize the system to be in a state with equal(ish) amount of both liquid and gas. Then we allow the system to cool down so that the liquid water is no longer boiling. Now the system sits at an equilibrium between liquid and gas states.

Now, when we put ice on top of the jar, the water vapor condenses and gets converted to liquid, pushing the equilibrium to the right. But this decreases the overall pressure in the system since fewer particles now occupy the volume above the liquid's surface. This is essentially the system trying to pull itself back towards the original equilibrium i.e. towards the left of the equation, which it does by making more water vapor i.e. boiling.

This reaction-like picture helps in visualizing the system better in some cases, so I tried to add it alongside the pressure dynamics scenario. You may be interested in Le Chatelier's principle if you prefer this.