As we know, Daenerys Targaryen has three dragons. So why did she
only take one dragon with her into battle against the Lannister forces in the Reach?
Was there any significance behind it?
Dragons aren't horses. You don't just go and jump on a dragon and assume it will serve you. They won't serve you, they won't follow you, they won't spare you unless you are their rider.
There is a very deep bond between a Dragon and its rider, traditionally formed when the two are in eggs/in mothers womb, like Dany felt that Rhaego was reaching out to her dragon eggs inside her womb. There are also instances of eggs being placed in cribs of Dragonlords like Maester Aemon, King Aegon V etc received. And then there are also instances of Dragonlords claiming Dragons centuries older than they were like prince Aemond claiming Vhagar. Closest thing to that bond would be the bond between a warg and his/her wolf.
Anyhow, once a bond is formed, it is for life. Doesn't matter who dies first, but the bond is broken only once one of the parties dies. Dragons are known to take more riders e.g. Vhagar was Queen Visenya's Dragon. After her, Laena Velaryon, great-grand-daughter of King Jaehaerys I rode it. After Laena died, Prince Aemond claimed it. Riders however apparently do not take other dragons, King Viserys I did not take another dragon after Balerion died, King Aegon II didn't take another dragon after Sunfyre, Aegon III didn't take another after death of Stormcloud (Although mostly because he saw his uncle's dragon eat his mother alive). The only instance of a rider taking another Dragon would be Rhaena Targaryen, daughter of Prince Daemon and Laena Velaryon. Her first hatchling died hours after its birth, she eventually bonded with a new dragon named Morning. But since her first hatchling died just hours after birth, it is impossible that she ever bonded with that hatchling so her real first dragon would be Morning.
Daenerys is different from your average Dragonrider. She's the mother of Dragons so no doubt she shares a bond of affection with all three but that is not the bond she shares with Drogon. Drogon is a part of her and she is a part of Drogon. The two are one, and one is two, if that makes sense.
Dragons are smart beasts and they can sense distress of their riders like Queen Helaena's dragon snapped her chains when the young Queen jumped from her window to her death. Something of that nature happened in Meereen I assume when all three were put to action but even then, Daenerys rode only Drogon.
There's however an alternative explanation for Meereen, the dragons are creatures of fire and blood. They are attracted to battle and carnage like carrion crows are attracted to corpses. They sensed a battle and they wanted in. So they came but Daenerys controlled only Drogon, not the other two.
No Dragonlord in history has ever controlled two dragons simultaneously. Aegon the Conqueror rode Balerion but he never dared mount Vhagar or Meraxes because they weren't bound to him and would have killed him.
From A Dance with Dragons:
It was said that even Aegon the Conquerer never dared mount Vhagar or Meraxes, nor did his sisters mount Balerion the Black Dread. Dragons live longer than men, some for hundreds of years, so Balerion had other riders after Aegon died... but no rider ever flew two dragons
Prince Joffrey Velaryon was killed by his mother's dragon Syrax when he thought he could use the she-dragon to get a lift to his own Dragon. Syrax let him get on top as she was used to his scent but once he started riding her, she threw him down to his death because she was bound to his mother, not him.
So Daenerys took only one Dragon to battle because that's the only one she could have taken. That's the only one she could control. Meereen was different, blood was nearby and Dragons don't need to be told what to do when a battle is raging. This was different, they were going thousands of miles away to attack Lannisters. She had no way of convincing the other two to follow her if they didn't want to. Of course the counter argument would be they followed her from Meereen to Dragonstone. But Dragonstone has special attraction for Dragonkind, think of a homing beacon. King Aegon II's dragon sunfyre found his way to Dragonstone despite being severely injured. Rose Road is an entirely different thing. In any case, there is precedent for one rider leading multiple dragons:
From The Rogue Prince:
In 116 AC, in the Free City of Pentos, Lady Laena gave birth to twin daughters, Daemon Targaryen’s first true-born children. The prince named the girls Baela (after his father) and Rhaena (after her mother). When they were half a year old, the girls and their mother sailed to Driftmark, whilst Daemon flew ahead with both dragons.
The counter-argument would be, dragons who feel natural attraction to Dragonstone were born and raised there. Some were raised in King's Landing but even they hatched in Dragonstone. Daenerys' Dragons were eggs from Asshai, hatched in Dothraki plains. They ought to feel no attraction to Dragonstone.
Plus with no riders, who was gonna ensure that Rhaegel and Viserion won't simply turn on their own soldiers? Without a rider, they are a liability and could have been killed more easily than one with a rider. The risk was too great.
SPOILERS for Episode 6
In episode 6, Daenerys takes all three dragons beyond the wall to fight against the dead and rescue Jon. She still rode only Drogon but she exhibited capability to take all three of them to war. So it seems, in show-verse, she didn't take all three just to minimize the risk of friendly fire and loss of dragons to some stray iron bolt. There might also have been some consideration to reduce casualties. Because more survivors means more recruits for Dany. Sure they would be of dubious loyalties but Aegon the Conqueror also used troops with dubious loyalties from his conquered foes against his still in-field foes.
She just lost the Tyrells mainly because she sent the Unsullied on the wrong battlefield and left her allies unprotected. She lost the Greyjoy and the dornishmen thanks to Euron. The Unsullied are far away from Dragonstone.
The Dothrakis are now fighting the Lannister's army so there is no army on Dragonstone, nobody to defend her fortress. It's for me the main reason why she left her two others dragons.
Now we might ask why she did not send her three dragons on the battlefield and left The Dothrakis on Dragonstone.