This is what I found on Wikipedia for h aspiré:
One part of the major phonological changes between Latin and Early Old French was the loss of the consonant /h/, which would later return with the introduction of Germanic words into the language. The aspirate h ceased to be pronounced once more in either the 16th or the 17th century, but some grammarians kept insisting for it be pronounced into the early part of the twentieth century.
Now, I checked the book that was cited as a reference but didn't manage to find the answer to the question as to why did the French stop pronouncing the h sound? Also does the bolded part mean that those words which are today pronounced with h aspiré were then pronounced with h muet?