Tom Bishop is a complete moron, he has no grasp of any scientific theories bar his own unique version of FE theory, not even the other FE believe him. We know the distance of stars from a variety of methods
Spectroscopy can provide a good indicator of a star's distance. Super giants have very different spectra from a normal hydrogen fusing star of the same spectral type. White dwarfs have very different spectra than either a giant, super giant or hydrogen fusing, also called Main Sequence stars. Each stage in a star's life stamps certain distinct traits in their light astronomers can find in spectra, and thus retrieve information on it's brightness, size and distance from Earth.
For stars that are relatively close, they use parallax to calculate the distance. I have to say, it never ceases to amaze me that this method actually works, since it requires measuring extremely small angles - we are talking arc-seconds, which are 1/3600 of a degree. To give you an idea how small an arc-second is, it is the angle the sun moves in 1/15 of a second due to the Earth's rotation. So, take a look at the sun for 1/15 of the second and let me know how much movement you notice.
To measure distances beyond 100 light-years, one method is to use Cepheid variable stars. These stars change in brightness over time, which allows astronomers to figure out the true brightness. Comparing the apparent brightness of the star to the true brightness allows the astronomer to calculate the distance to the star. This method has been used to find the distances to many globular clusters. Red shift is another trick, and it is used to measure the distances to really far away objects such as other galaxies.
By parallax, what we do is snap a photo of the starfield, say in January. Then we do it again in July, as the earth has swung halfway around the sun. Knowing that the earth is 93 million miles from the sun gives us the base of a triangle about 186 million miles long. Trivially small, in comparison to the distance to the nearest star, but you can get a reasonably fair estimate this way.
For example, be estimated the distance to the Large Magellanic Cloud. By the 1980s we had improved our accuracy and we estimated the cloud o be between 150 and 200 thousand light years. Then a supernova of the star Sanduleak -69 202 within the Magellanic Cloud. After it blew, it lit up a surrounding ring six light months away. Well, now we have a triangle with a base and an angle (we could measure the degrees between the star and ring). The distance pops out to 170 light years.
Knowing accurately how far away that is, we get better numbers for Andromeda (2.5 million ly), and everything else.
Standard candles like cepheid variables and redshifts are still necessary for distant galaxies, but those distance estimates are based on accurate close measurements.
Another trick for "nearby" stars is to know the distance between orbitting binaries. You can measure angles that way as well