I have forgotten how scientists measure the distance to stars by studying nearby varible stars. Please explain.

A spectrum analysis of the light from the star (i.e. redshift) can determine how long the light took to get from the star to Earth, and since the speed of light is a known constant (3^8 meters/sec.) that can be used to determine how far the light traveled.

There is a certain type of variable star known as a "Cephiud," which is believed to always shine at the same intrinsic brightness. It is therefore a convenient "standard candle." Astronomers look for these particular star types in other galaxies, measure the star's apparent brightness, and then determine the distance to that star based on the fact that the brightness is proportional to the inverse square of its distance (i.e. a star that is twce as far away appears 1/4 as bright). Measuring these variable stars is what leads to an estimate of the value of Hubble's constant, which is what Adrastus seems to be referring to.