How does a star like the sun become a red giant?

The sun, like other stars, is dependent on nuclear reactions for its energy. The particular type of nuclear reaction that powers the stars is nuclear fusion, in which smaller nuclei combine to form larger nuclei. Specifically, for a star in the main sequence (just like our sun), hydrogen is being converted to helium during the nuclear fusion process. Over time, the hydrogen at the core will be used up. When this happens, the core contracts as the fusion of hydrogen to helium stops and the fusion of helium to carbon starts. However, the outer portions of the star still contain hydrogen and the fusion of hydrogen to helium goes on in there. This causes an expansion of the outer portions of the star and the star grows in size. This also causes a cooling of the star and it turns red (and that is why it is called a red giant). 

All stars become a red giant at some point in their lifecycle.

Hope this helps.