The entire matter is detailed in Robert Caro's second book in his biographical series on Johnson. The book is entitled Means of Ascent.
Johnson election theft took place in 1948, when he was running for the Democratic nomination for U.S. Senate against Texas Governor Coke Stevenson, one of the most admired and respected governors in the history of the state.
In the primary election, Stevenson led Johnson by 70,000 votes, but because he didn't have a majority of the votes, he was forced into a run-off. The run-off was held on a Saturday. On the Sunday morning after the run-off, Stevenson was leading by 854 votes.
As a New York Times review of Caro's account stated, the day after the run-off election it was "discovered" that the returns of a particular county had not yet been counted. The newly discovered votes were overwhelmingly in favor of Johnson. Then, on Monday more returns came in from the Rio Grande Valley.
Nonetheless, on Tuesday, the State Election Bureau announced that Stevenson had won by 349 votes. Nothing changed on Wednesday and Thursday after the election. On Friday, precincts in the Rio Grande Valley made "corrections" to their tallies, which narrowed Stevenson's lead to 157.
But also on Friday, Jim Wells County, which was governed as a personal fiefdom by a powerful South Texas rancher named George Parr, filed "amended" returns for what has become famous as "Box 13" that gave Johnson another 200 votes. When all was said and done, Johnson had "won" the election by 87 votes.
It was later discovered that one of Parr's men had changed the total tally for Johnson from 765 to 965 by simply curling the 7 into a 9.