The riding hall really is for the lazy people among us ;) You click the button and 4 hours later, the pet has one more level.
The advantage is that you can use it to level up any pet, (like for example your weak gryphon) and you can even use it on a pet that doesn't have any Energy left, but it takes 4 hours (during which as you noticed the energy doesn't change and the pet can't do anything else) and it costs silver.
The cost is dependant on how much exp is needed to next level, so a Level 1 that you want to get to level 2 will cost very little (80s), but higher levels are a different matter.A few examples from my own stable: a level 46 with 0 Exp will cost 3680s. A level 48 with 54% exp 1768s.
It is still useful if you have many pets that you want to level and you can't go exploring or battling with them all. Also what I like to do in the evening, I battle or explore my pets to 0 Energy and then if they are fairly close to a level, I put them in the training hall so they level up while I sleep.
If you have additional questions, feel free to ask, but I hope this covers everything about the riding hall ;)
