The static electric field inherent within all diodes (and therefore LEDs) along with phonon (heat) interactions with the lattice is what causes reverse leakage current. LEDs are generally diodes with specific band-gaps so from here on, I'll just talk about diodes.
Diodes are pn junctions. A pn junction in the simplest sense is a collection of static positive charges on one side and a collection of static negative charges on the other side. In this small region between these static negative charges and static positive charges is an area where if an electron-hole pair is created, the electron will head towards the positive static charges and the hole will head toward the negative static charges. This would create a small current if we connected the terminals of the device together.
There's a couple things that are well known to create electron-hole pairs in a lattice.
1) photons hitting electrons exciting them to higher energy states across the band gap.
2) phonons (waves caused by heat) hitting electrons exciting them to higher energy states across the band gap.
Photons are generally much more energetic so we utilize this phenomenon called the photoelectric effect in solar cells. Phonons on the other hand tend to be of much lower energy so you only get a minuscule amount of current because most of them simply don't have enough energy to breach the bandgap of the semiconductor. With higher temperature, the band-gap narrows and the energy of the phonons increases which will increase your reverse leakage current.
If you wanted to decrease this current, you could do three things, increase the band gap you're using, cool your device down, or decrease the width of your pn junction. In all cases, you'll reduce the amount of electron-hole pairs being generated within the region which will reduce your current.