To expand your inventory, you need to equip backpacks. Your Character can equip up to 4 of those.
You get a free backpack in the very beginning of the game. Apart of that you will either have to use AP, which you can get by spending real money, to buy one from our Item Mall
or you save up money to buy them ingame. A 5 Slot backpack is around 400-500 Gold ingame, at least it was that much when I checked it last time.
There are various sizes of backpacks: 5, 7, 10, 15 and 20 Slot backpacks. If you want to get anything that has 10 or more slots, you will need to fuse several 5 slot bags, so it is pretty expensive to get there.