How do I re-mount my horse?

No, currently remounting is not possible. As OP mentioned in their answer, remounting was restricted to the closed beta.

However some theories have been put forward as to why remounting was removed by the time the game was released. Some claim that remounts used to be a necessity back when maps were gigantic, and using a mount was critical in traversing the map. However once the game was released, the developers realized that walking did not take too long, and so remounting was no longer necessary.

Others (see comment #10 in thread above) point out the possibility of someone using their mount to quickly enter battle, popping their ult, and then remounting and escaping - something which was unfair, and thus prevented by preventing remounts altogether.

To address your question more directly, once you leave spawn and dismount your horse, there is no way to remount it. However, as someone pointed out in the comments to your answer, your dismount could have been as a result of being hit by an enemy. Either way, remounting is currently not possible without cheating or hacking, which will get you banned from the game.

There is currently no way to get back on your horse once you have dismounted.

There was a remount button during the game's closed beta, but it was removed before the game's public release.