The steam doesn't leave, so heat and pressure doesn't leave. It's like how you get warmer with a blanket on

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The lid prevents convective cooling, which is the cooling from air movement. Without the lid, the hot air at the surface of the water rises, taking some of the heat with it, which is then replaced with cooler air from the room. So you have to keep heating the air over and over when the lid is not on.

The lid retains heat and pressure inside the pot, which causes the water to boil more quickly than if the surface were touching room temp air and the pressure didn't build.

The pressure leaves immediately, and so does necessarily some steam. Otherwise that pot would turn into a bomb very very fast. Actually very little pressure is needed to lift the lid up; one atmosphere of pressure is some hundreds of kilograms on a decently sized one, so maybe 1% of that is enough.

The lid prevents air convection that carries the hot and especially wet air away. While as one answer says the resulting circulation of air cools the water, this is not the main factor. Instead, it is evaporation. Water evaporates all the time, and the closer it gets to boiling, the faster it gets.

Water uses a lot of energy to become steam, way more than it takes to get it to boiling temperatures. Every bit of steam that escapes is enough to heat way more than this amount of water all to boiling. So you want to prevent evaporation. but the speed of evaporation depends on the wetness of the air above the water. Hence a lid, to keep enough wet air inside.

And as a bonus, it prevents circulation, which would also carry away wet & hot air. Lastly, there is also a bit of energy recovered when water condenses on the lid. Condensation releases just as much energy as vaporizing takes. That effectively improves the total insulation, meaning that less heat escapes to do whatever else outside the pot.

The pressure leaves immediately, and so does necessarily some steam. Otherwise that pot would turn into a bomb very very fast. Actually very little pressure is needed to lift the lid up; one atmosphere of pressure is some hundreds of kilograms on a decently sized one, so maybe 1% of that is enough.

Probably more accurate to say that steam produced under the lid will tend to condense on the underside of the lid and eventually run down back into the pot. It is transferring some heat to the lid in doing this, but the lid can’t radiate heat away as quickly as steam can take it away when it drifts off.

Why do things cook faster in the oven when the door is closed?

Mainly because the lid reduces evaporation. Water evaporates all the time, not just when it's at boiling temperature. Or rather, it evaporates if the air is able to accept more water vapor. Air that is already very humid can't accept much more water. At 100% air humidity, no (net) evaporation takes place.

Okay, so why is this important to how fast water boils? Well, when water evaporates, it steals a lot of heat from its surroundings. That's how sweating cools down your body: the sweat evaporates off your skin and cools it. So, if you're trying to heat up a pot of water, evaporation is your enemy.

If you leave the pot uncovered, evaporation can happen freely as humid air above the pot is replaced by drier air. But if you cover the pot, now air gets replaced much more slowly (it will still get replaced a little). Water will evaporate into the air above it, but this air will quickly reach 100% humidity and then no longer be able to accept any more water vapor. This doesn't mean that evaporation halts completely, but rather an equilibrium is reached where some new water evaporates, but an equal amount of existing water vapor condenses against the walls and (especially) the lid of the pot. When water vapor condenses, the energy it "stole" before to evaporate is returned to the area where the condensation occurs. So, if you can get the vapor to condense inside the pot, then the heat stays inside the pot.

A smaller effect is that the lid also traps heat. Air above the water heats up, and if the pot is uncovered, this hot air is replaced by cooler air, which cools down (or slows down the heating of) the pot and the water. Trapping the hot air above the water means you again lose less heat, but the reduced evaporation will always be the larger effect.