Why do stars become red giants?
(This is somewhat simplified but I hope it gets the idea across.)

The reactions stop in the core because it runs out of fuel. During the main sequence, the star is supported by the fusion of hydrogen into helium. Eventually, the hydrogen runs out at the centre, so hydrogen fusion is no longer possible there.

Why doesn't it start fusing helium into carbon right away? That's because the core isn't hot or dense enough yet. Different reactions broadly rely on the presence of different resonant states in the nuclei and, in the case of helium, such a state cannot be reached often enough until the core temperature is about $10^8$ kelvin.

In order to get that hot, the core has to contract and heat up. It eventually does (if the star is massive enough) but it doesn't happen instantaneously. Remember that the gas is still hot and at high pressure, which it exerts on itself and its surroundings.

Meanwhile, at the edge of the core, the star (partly as a result of said contraction) is hot enough to turn hydrogen into helium, so it does so. This is exactly the nuclear-burning shell that distinguishes the internal structure of a red giant.

So maybe think of it this way. Imagine a star at the end of the main sequence. Where is it hot enough to fuse hydrogen into helium? Everywhere up to the edge of the core! Does it fuse in the core? No, because it's out of fuel. So where does it fuse? At the edge of the core, which we recognize as the shell.