Leaguestones are only found in the Legacy League, so you need a character in either Legacy League to be able to get them. Generally they can drop from any enemy just like all other items, and blue and yellow mobs also have a higher chance to drop them.
You will begin finding Leaguestones already at the beginning of the game, but they will be tied to a specific level bracket, so it is no need to save them until endgame.
To use them, simply right click on them while they are in your inventory. This will equip it onto your active Leaguestones (which can be accessed by clicking on H). You can have up to 3 active leaguestones at a time, and you can easily remove them from this interface the same way, if you don't want to use it now.
Each Leaguestone generally comes with 5 charges, so it will trigger in the first 5 zones you enter. Some Leaguestones have extra charges, and there is no way to increase the charges for the stones at the moment.
You can also buy and sell Leaguestones like any other item, and this can be done by going to poe.trade and search for the Leaguestone you are interested in.