I really encourage interested folks to read the biography (though it's an undertaking).
According to Caro, part of the background is that the relevant southern Texas precincts were well understood to have vote counts up for purchase; over the course of election counting, both sides would have their controlled districts release counts based on what the other side was reporting to stay in the race. These counts would vary in legitimacy and how skewed they were based on the precinct and need of the candidate that had swayed the boss to their side. But tactics like having armed guards supervise the casting of votes to ensure the favored candidate got a large majority, or simply distributing vote receipts to people who never voted at all and recording votes on their behalf, or making numbers up entirely, were quite common. Typically, though, Caro argues that because both sides did this, and they did it incrementally, it usually wasn't enough to sway an election one way or another, but rather was just part of the cost of doing business. He even says that LBJ lost his Senate election earlier that decade because he got cocky and told the bosses of the districts he had bought to just release all their numbers right away, letting his opposition then juice their numbers just enough to win.
It's really the timing, more than the margin, that makes it clear what happened (and the crudeness of the forgery); after every other precinct reported and finalized, they corrected their number by barely more than needed to win. The 100 to 1 vote margin was actually not that far off from the vote margin that the precinct reported in the first place (... which, of course, really tells you that the whole thing was made up from whole cloth).