Storms and cyclones both begin with warm air rising and leaving a low-pressure region behind. A cyclone, however, also needs the Earth's rotation to make that inrushing air spin — so if the Earth stopped rotating, cyclones as we know them would not form.
How a storm forms: the ground (or the sea) heats up, the warm and moist air above it becomes lighter and rises, creating a low-pressure area. Cooler air from surrounding high-pressure areas flows in, gets heated and rises in turn, so a continuous wind circulation is set up. The rising air expands and cools, its moisture condenses into water droplets and clouds form; the droplets merge into heavier drops that fall as rain, hail or snow. Strong winds together with this rain make a storm.
How a cyclone forms: over warm ocean water the same rising continues, but each time water vapour condenses into raindrops it releases the heat that was taken up during evaporation. That heat warms the ascending air further, so it rises even more, and the pressure at the centre falls even lower. Air from all around rushes in and begins rising too. The Earth's rotation makes this moving air spin, and the cycle repeats until there is a very low-pressure centre with high-speed winds revolving around it — a cyclone.
Why it happens: the spin is not produced by the storm itself. The book says plainly that Earth's rotation causes the moving air to spin. On a non-rotating Earth, air would still rush straight in towards a low-pressure centre and there would still be heavy rain and strong winds — but the inflowing air would have no reason to curve, so it could not organise into a rotating system with a calm eye at its centre.