I will try and help you since I just spent little over 68 hours of playtime on this game. x.x And rising! Rofl
The ones that looks like Pyramids are actually a grid meaning the stage so for example: On The Ground> When your on the ground.
You can be fighting running around, and you will gain 1.2x exp for doing so.
'i' are accessories that goes in to effect depending what your opponent is in.
'EX' is being in EX mode, whether it is you or your opponent you'll gain experience. Example: EX Mode> 'While you are in EX Mode'
Basically, when you enter EX Mode you will gain 1.3x exp for being in EX Mode.
'HP' depending what it says if it's, 'HP = 100%', 'HP is 50-70%', etc. You'll gain a set amount of exp for having that amount of HP. Example: Near Death> 'When you are near death'
So when your low on HP and your character is red, this will come in to effect, and if you manage to win in the 'near death' state you will earn 1.5x exp.
'BRV' accessories will reward you with exp depending whatever you are going to put on. Example: Break> 'When opponent is suffering from Break'.
So when you break their bravery points the icon will light up (if equipped) and you'll gain 1.5x exp if you defeat him in that state.
The icons with Red stones that looks like the summonstones, are simple and sometimes difficult if you don't have the right summon on Auto or Manual (R+O). Example: After Summon> 'Once you have used a summon'
After you have used a summon you will gain 1.5x exp.
Swords crossed icons vary in whatever they maybe, for the easiest understanding of this would be the 'Attacking Bravery' and 'Attacking HP'. You'll gain 1.2x exp for either attacking BRV or HP (depends what you choose to equip.)
Clock icons are simple, pretty much if one says '30 seconds after battle starts'. You can take advantage of knocking the opponent's Bravery out and/or running around destroying the stage. Pretty much whatever you decide to do that won't kill him before 30 seconds has passed in the match. It will reward you 1.2x exp.
I hope this helped you understand what those icons mean and have an idea on what some of them can do to help you level as well make you stronger. If anyone can explain this better than I, sorry for lacking in some teaching skills. x_x