Much like increasing health, increasing stamina involves finding hidden collectibles. This How To Increase Grip Gauge (Stamina) In Shadow Of The Colossus will tell you what you need to do to increase your stamina and hold on to Colossus enemies for longer.
Your grip gauge is very important as it represents how long you are able to cling on to a Colossus when they are attempting to shake you off. Basically, any time you are holding the R2 button your grip gauge is being drained.
There are two main methods of increasing your grip gauge. Firstly there are Colossus battles. Each time you defeat a Colossus you get a slight bonus to your grip gauge. Secondly, collecting lizards. There are special lizards with blue tails scattered throughout the map. You must shoot the lizard, kill them and then walk up to them with R2 to eat their tails. This increases your grip gauge. They are very hard to spot though.
For future reference, check out the main temple area. Inside the temple, look up to the large dome type part that has rays of sun shining through. There is a lizard on one of the sizes within the dome.