A couple of possibilities exist. Planets close to the star are engulfed in the expanding star, spiral inside it, and are destroyed. In our own solar system, Mercury and Venus are doomed. As the star expands to form a red giant, it also starts losing mass.
All stars lose mass.
For instance, the Sun is losing mass. When a star enters the red giant phase, the rate at which it loses mass can accelerate. The mass of a star determines how far a planet orbits from it. Thus, as the Sun loses mass, the orbits of the other planets will expand. The orbit of Mars will almost certainly expand faster than the Sun does, thus Mars will probably not suffer the same fate as Mercury and Venus.