Possible causes for AAA battery to explode
From the information provided it sounds either like a faulty battery or a very high current drain from the equipment - or both.



Update 1 : Having seen the most impressive photo, my prior assessment stands. This would be extremely unusual. A large amount of energy seems to have been involved. If there were 3 or more batteries in series (were there?) and one was reversed this may happen as the current would be driven through it backwards. 

This strongly suggests a  bad battery - possibly a counterfeit one. 





Update 2:

We now know there are two batteries.
 This is less than the 3 minimum needed to drive current backwards through one battery so the back discharge mode seems unlikely.
 it is still possible with one well charged battery and one fully discharged.
 The good battery can effectively reverse polarity charge the dead battery.
 Unlikely but possible in this case.

A counterfeit battery still sounds possible.



Prior material:

IR remote controls pulse IR LEDs with short pulses of very high current - possibly an amp or more. Most batteries should either provide this or just gracefully fail to do so. A very poor quality battery or a faulty one MAY be affected by such a load.

If the IR control stayed on for some reason then a continued high current may occur. If this happens the IR LED would probably die. If your control still works with a nw battery then this is probably NOT what happened. 

It is EXTREMELY unusual for an AAA cell to "explode" in use. You need to say if it was an alkaline, or NimH or ??? type cell.

Some appliances allow charging of the battery inside the equipment. If a non rechargeable battery is charged it MAY explode. This would be rare and it is unlikely your remote allowed charging. 



Aspects worth considering in situations like this:

These are suggestions only - necessarily an incomplete list.

What sort of battery -  Alkaline, NimH, other?
Ability to deliver high current may increase chances of "energetic" reaction. 

How many AA batteries in remote?
Three batteries are required in series for reversal of one battery to cause significant reverse current flow when all batteries are in good condition. (ie one "forward" battery opposes the reversed battery and the remaining 'forward' battery supplies forward current. 

IR remote presumably?
IR remotes often pulse the LEDs at very high peak current levels - far higher than in most handheld devices. 
Brand of battery?

Age of battery (time in use)

Was it recharged 

Was a non-rechargable battery charged? Can 'cause problems'.