The method to upgrading your inventory size in Forager is to upgrade your backpack. To do this, you must first unlock the Sewing Station, a building that is essential for crafting a lot of different materials and tool upgrades. You should invest a Skill Point into unlocking this pretty early in the game, and build one when you have enough resources for it.
|There are lots of different items to loot in Forager.|
When you have the Sewing Station you will be able to craft a "Small Backback" for 4x Thread and 2x Flowers. This should be easy to get your hands on, even if you only have the starter Grass Biome unlocked. Crafting this will give you a permanent +4 inventory space!
After you have crafted the Small Backpack and gotten the +4 inventory spaces, a new upgrade is available. Each new backpack upgrade gives you yet another +4 permanent inventory spaces, but they also requires new and rares materials for each additional upgrade.
After having bough all the upgrades, you will have a total of 20 inventory spaces, which still might not be enough. If that's the case, make a few Vaults and store the items you don't immediately need there. The Vaults are way more powerful than a lot of people think, since all new items you get will get automatically stored there after you have placed an item there. This means that if you put your Flowers in the Vault, all the new Flowers you loot will be automatically transferred to your Vault, and won't ever bother you with taking up an inventory space until you go to the Vault to get it.
Anyway, that's all you need to know about increasing your inventory size in Forager, since this is the only method to doing so. Enjoy your upgraded backpack!