Publius has got it
Even Michigan State used to be the Michigan A&M Aggies (and their yearbook was called "The Wolverind")
"Aggie" used to be a mild pejorative towards agricultural workers, just like "techie" or "geek" is today towards, well, geeks. So "Texas Aggies" was as much to distinguish us from, say, Oklahoma or Michigan Aggies, as it was to distinguish us from the generic lot of aggies. And aggie jokes are not originally about student from our school; they were about aggies in general. These days jokes about agricultural workers are as anachronistic as jokes about "Pollocks." We Texas Aggies are the last best and baddest group of Aggies standing, so we inherit the status of being the butt of "Aggie Jokes." It's a good thing.
You can look at what the government of India did with its IIT system for an analogy to what the US government did right after the Civil War with the land grant schools. And, it's not too hard to imagine that one of the IIT schools might, cheekily, call themselves "The Bangalore Nerds" or something like that. Well, that's why there are so many Aggie schools.
Another note is that there was as second land grant act, passed in 1895 I think, which set up a sort of "separate but definitely not equal" system of agriculture-based schools to educate the black population (viz. former slaves). In many ways this initiative was modeled on the original land grant act, so the mascot of Aggies was likewise widely adopted (e.g. NC A&T, Alabama A&M).
Now, why we are one of a very small number of remaining land grant schools (from either land grant act) holding on to "Aggies" is a different question -- that question is more along the lines of "why so few other Aggies?" I don't know the answer to that, but I can say that it's at least coherent with Texas A&M's established conservative character.