How does a star become a red supergiant?
Large stars become red supergiants when they run out of Hydrogen in their core.
Red supergiants start out an main sequence stars with masses between about 10 and 30 times that of our Sun.
When the fusion process exhausts the supply of Hydrogen in the core, the core is mainly Helium. The area around the core is mainly Hydrogen and it is hot enough to start fusing. This fusion reaction causes the outer layers of the star to expand rapidly.
The expansion process causes the outer layers to cool to under about 4,000K and hence have a red colour. They are very large, typically hundreds to over a thousand times as large as the Sun.