All kinds of speculation is circling around this very odd behavior. Reports National Review about the incident:
[H]is Thursday remarks clocked in at less than 20 minutes...I'm hearing speculation that Senate Majority leader Mitch McConnell is getting sick and tired of Rand's antics and Rand's refusal to quit the presidential race to focus on his Kentucky senate re-election bid, and that Mitchell ordered Rand off the floor.
A few minutes later, Paul was gone, to the confusion of his colleagues. “We’re all kind of a little shocked,” a Senate GOP aide tells National Review.
Others speculate that Rand may be suffering from something close to a mental breakdown, given his poor performance in the polls and the unceasing harshness of media commentary about his campaign and alleged prickly personality.
Whatever caused Rand to leave the floor of the Senate, his campaign is certainly not going to publicly state. They seem as confused as everyone.
[After Rand walked off] Eleanor May, the national press secretary for Paul’s presidential campaign, says that “he will return to the floor around one a.m., once session resumes.”That didn't happen.
The next explanation didn't make any sense either, as NR points out:
His team told ABC that he didn’t continue speaking because Senate majority leader Mitch McConnell (R., Ky.) had already filed cloture, the procedural move to end debate on the debt limit; it’s an odd answer, because that took place on Tuesday, before Paul touted his coming filibuster at the presidential debate.
With cloture filed, the earliest that the Senate could vote on the debt-limit agreement is 1:00 a.m. Friday. If Paul had refused to allow the Senate to adjourn for the day, he could have theoretically delayed that vote to 1:00 p.m. Friday. “It’s meaningful, its a real delay, but he can’t prevent it from happening [entirely],” as the GOP Senate aide put it.