Why Helium-burning Stars are found in a Horizontal Branch? Why Helium-burning stars are found in a horizontal branch on the H-R diagram? Why do they have roughly the same luminosity?
This is explained in the Wikipedia article 


  Stars on the horizontal branch all have very similar core masses, following the helium flash. This means that they have very similar luminosities, and on a Hertzsprung–Russell diagram plotted by visual magnitude the branch is horizontal.


To expand a little. In stars of a certain mass range, helium builds up in the core until it reaches a specific mass, at which point the "Helium flash" occurs and burning of helium to carbon and oxygen starts throughout the core. When things settle down, helium burning is going on in the core, which is more or less the same size, independently of the original mass of the star. Since this is the main power source of these stars, they all have about the same luminosity.

The variation across the branch comes from how much remaining gas there is outside the helium-burning shell -- more gas means a larger cooler star radiating the same total amount of energy