To be fair, the “h” was silent in UK English when the English colonized America, and then the English began pronouncing the “h” at a later point. UK English deviates from the common ancestor surprisingly frequently.
I remember a flame war between US-English and UK-English speakers, where one of the former said “if it weren't for us, you'd be speaking German now” and one of the latter ones said “if it weren't for the French, you'd still speak English now”.
I don’t understand the rebuttal. Is American English supposed to be more French-influenced than UK English?
It is a reference to the assistance provided by France to help the American colonies (and later the United States) rebel against the British Empire.
Ah. That's not a very sharp barb considering the significant power differential between England and the Revolutionists at the time.
I think the key part of the joke is that Americans don't speak proper English.
Like random3821 commented, it's about American English not being “proper” English (while supposedly if the US of A were a British colony, it would be “proper”.)