I have studied that the principle of equal a priori probabilities yields maximum entropy principle and minimum free energy principle and we can define and calculate other thermodynamic variables. However, where does the principle of equal a priori probabilities come from?
Principles ( and laws, and postulates) in physics are the equivalent of axioms in a mathematical theory. The mathematical format used to study physics is very broad . A subset of the possible solutions allowed by the mathematical axioms is picked up by the use of laws, as the conservation laws, and principles, as for example the least action principle, to formulate models which can fit existing data, and, very important, be predictive of future data.
Physics, in contrast to mathematics, does not end with "quod erat demonstrandum". The models have to be validated or falsified by data.
Principles can be formulated from observations. The simplest and most studied by all probability distribution is the throw of the dice. The equal probability for all the faces is important for the dice to be unbiased, and this is an observation based on the randomness of the throws and the weight uniformity of the dice.
For dice it is easy to "prove" that if the throw is random and the dice matter uniform the probability distribution will be flat. It is not so for many particle systems treated in statistical mechanics, discussed here . Assuming it as a principle allows for picking the mathematical formats used in modeling many particle systems with statistical mechanics.