From the perspective of 19th-century cosmology (and before), the universe was infinite, unchanging, homogeneous, and therefore filled with stars.
The faulty premise, unknown in Olbers' time, was that the universe is not infinite, static, and homogeneous.
The Big Bang cosmology replaced this model (expanding, finite, and inhomogeneous universe).
One of at least several explanations is that distant stars and galaxies are red shifted, which weakens their apparent light and makes the night sky dark.
which make the description of the evolution of the system depend upon its position (potential wells, etc.).
This only stems from the fact that the objects creating these external fields are not considered as (a "dynamical") part of the system.
This is shown, using variational calculus, in standard textbooks like the classical reference text of Landau & Lifshitz.
As said in the introduction, dimensional homogeneity is the quality of an equation having quantities of same units on both sides.
A valid equation in physics must be homogeneous, since equality cannot apply between quantities of different nature.
For example, the following formulae could be valid expressions for some energy: if m is a mass, v and c are velocities, p is a momentum, h is the Planck constant, λ a length.
Being homogeneous does not necessarily mean the equation will be true, since it does not take into account numerical factors.
For example, E = mv2 could be or could not be the correct formula for the energy of a particle of mass m traveling at speed v, and one cannot know if hc/λ should be divided or multiplied by 2π.