What we’re missing is a phenomenon that probably was described correctly by your high school science teacher, namely, the phenomenon of “latent heat” (which is what I was taught, though “enthalpy” would be the more modern term.)
I said that the temperature of a substance is the average amount of energy in it. I lied; it is possible to add energy to water without changing the temperature. Liquid water requires “extra” energy to vaporize that is not reflected in the temperature of the water. You may recall doing a science experiment which ended up with a graph like this one.
As you can see from the graph, when the water reaches the point where it is about to melt or boil the temperature stops going up for a while, even though more and more energy is being added to the mix. The liquid water requires an extra little boost of energy to liberate the molecule from the liquid. That amount of energy is roughly 540 calories per gram of water thusly liberated.
Let me be very clear on two points here. First, liquid water evaporates at any temperature, not just the boiling point. Obviously puddles dry up even though they are not boiling! The boiling point is simply the point at which the liquid water has so much heat, and the vapor pressure is sufficiently low, that all of it is going to rapidly turn into vapor in short order. And second, with that in mind it should be clear that the 540 calories required per gram of vaporized water is 540 calories per gram no matter what temperature the water is at when it is vaporizing. (There may be some small variation in the latent heat required based on the temperature, but it'll be on the near order of 540 calories per gram in the kind of temperature ranges we're talking about here.)
One way to think about the latent heat is this: liquid water has way more entropy – more randomness – than solid water. The position and speed of each molecule in a liquid is way more random than in a solid. And similarly, water vapor has way more entropy than liquid water. Adding heat makes entropy; it disorders a structure. If you want to make water more entropic by evaporating it, you’re going to have to add a whole load of heat to it to account for the increase in entropy. 540 calories per gram, in fact.
But this process goes the other way too. When you export entropy from water vapor by turning it back into a liquid, those 540 calories per gram have to go somewhere. That entropy is exported from the water in the form of heat. When you put your finger in the steam rising from a boiling kettle, it's the enormous latent heat of the vapor condensing on your relatively cold finger that really scalds you. Even if you happened to be in atmospheric conditions where the condensation happened to a much lower-temperature vapor, the latent heat would still hurt a lot.
So now let’s revisit our look at a nice fluffy cloud. The sun adds 540 calories to a gram of surface water, liberating it into the atmosphere. That gram of water winds its way up into the sky where it hits a region of sufficiently low temperature and high pressure as that it flashes back into liquid form. Without changing the temperature of the water further, 540 calories is suddenly liberated into the bottom of the cloud, heating up the surrounding air. Hot air rises. An updraft forms in the middle of the cloud, sucking some of the water droplets higher into the atmosphere. The cloud gets taller. Eventually the updraft cools off and the cooler air moves outwards and slowly subsides. You'll get a sort of doughnut-shaped circulation of air around the exterior of the cloud as the falling air then gets sucked back in the bottom.
Meanwhile, the bottom of the cloud is getting hotter and hotter from the latent heat of more water evaporated from the lake surface turning back into liquid. The bottom of the cloud has an updraft in it, so it is pulling more and more water vapor from the wet atmosphere below it into the bottom of the cloud. The process is accelerating. If the cloud happens to be over a huge body of warm water then this can accelerate very rapidly.
The latent heat of fusion liberated when a gram of liquid water turns into ice is another 333 calories on top of the 540 calories that was from the latent heat of condensation. Imagine what happens to our cloud if the strong updraft makes the cloud so tall that the extremely cold atmosphere up there glaciates the lifted droplets into a cirrus cloud at the top. Cirrus clouds are made out of ice crystals. The water (now in the form of ice) escapes from the cloud out the top and potentially stays up there for a long time. The ice drifts slowly away from the top of what is now probably looking like a pretty threatening cloud. Essentially the cirrus cloud is extracting all the latent heat from the water, depositing the heat in the cloud, and exporting the resulting coldness out the top in the form of ice.
Now imagine what happens if a bit of the water vapor in the cloud gets cold enough to freeze in the middle of the cloud. It liberates its latent heat of fusion into the updraft, making it stronger. The updraft sucks the ice fragment up through hundreds of meters of cold wetness. When it gets to the top, maybe it is heavy enough to fall back down through more cold wetness in the outer part of the cloud, only to get sucked up by the updraft again when it reaches the bottom. The ice fragment gets bigger and bigger, liberating more and more energy into the updraft as it goes around in vertical circles. Eventually the updraft is not strong enough to hold it up and all that cold gets exported out the bottom of the cloud in the form of hail.
And that is how thunderclouds concentrate massive amounts of energy; they move huge quantities of latent heat that the sun puts into warm surface water into their interiors to power the very updrafts that suck in more vapor from below, concentrating the heat even more. The average heat keeps going up because coldness is exported in the form of cirrus clouds out the top and hail (or cold rain) out the bottom.
Now imagine the effect of hundreds of such clouds forming over a relatively small area. The suction from the updrafts is immense, and easily moves around millions of tonnes of air that we then experience as high winds. And when the updrafts are insufficient to hold the water up, you get torrential rains and hail falling out of the sky. It’s an amazing phenomenon, and it is all thanks to the fact that our planet is almost entirely covered by a substance which has high latent heat and can be solid, liquid and gas in a relatively small temperature and pressure range.