Great question! So this actually has a pretty straightforward answer via primary sources from the era. The standard protocol for dismounted cavalry deployed in a skirmish line was that every 4th man held the reins of his horse and three others. So in a body of 100 cavalry troopers, 75 would be in line to fire, while 25 would be slightly behind the lines tending to the horses. Naturally, this was a flexible command strategy, and when getting especially aggressive like Gen. Buford was on the 1st day, you might send some horses further behind the lines to get tied up and secured to free up more rifles. This was not the standard practice, however. The thinking behind the three-man fight, one-man-hold strategy was to keep the horses nearby and ready for rapid redeployment or retreat (sending the horses further back to free up more rifles negates this advantage).

Remember, during the American Civil War, the cavalry's primary use was to screen your own army's movements, locate the displacements of your enemy, pursue scattered bodies of infantry to exploit potential breakthroughs, and lastly, to conduct raids behind the lines against poorly organized defenses. You weren't supposed to give up a fighting unit's distinct advantage (horse power and speed) to tangle with simple infantry. Dismounting cavalry to fight against infantry the way Buford did on July 1st was bold, and is a credit to the Federal commander who saw the tactical advantages available to the Union if they could hold off Gen. Heth long enough for Gen. Reynolds to arrive and dig in. Most cavalry commanders would have extended skirmishers at best to draw out the Confederate's main bodies, then fallen back to report on the advancing enemy forces. Buford's tenacity and willingness to sacrifice parts of his command to buy time is one of the biggest reasons the Union was ultimately victorious, and was/is a brilliant study in how to use dismounted cavalry.

This is like a boxer or MMA fighter giving up a speed advantage to slug it out with a hard-hitting brawler. The fast fighter would normally dance around never letting themselves get in range of the bigger, more powerful foe. What Buford did on the 1st day of Gettysburg was get into a slugging match with a tougher, bigger, harder-hitting foe. He knew that he wasn't fighting to win, though, just to get to a full fight decision where he knew he'd be victorious (to extend the analogy, let's say our faster fighter made it through the last round, trading body shots, but instead of a decision, a new fighter is swapped in for each side, and the fight continues). The smaller, faster fighter didn't need to win, they just needed to survive long enough for their back-up to arrive and take over.

Interestingly, if you really want to read up on dismounted cavalry tactics and how the 3-man-fight, 1-man-hold strategy played out in an especially notorious situation, read up on Custer at Little Big Horn. Custer and Reno were also adhering to this dismounted tactic, and while it probably didn't make much of a difference, the fact that this was in play during the battle has been repeatedly observed in that historiography.

[Sources: Brent Nosworthy, 'The Bloody Crucible of Courage'; Carter Samuel III, 'The Last Cavaliers'; Stephen W. Sears, 'Gettysburg']