How do I increase my storage capacity?

In NieR:Automata, you equip chips to your memory, which improve you in some way, whether it be through HUD enhancements or attack/defense buffs.

However, I noticed that you start the game with less than what seems to be half of the maximum capacity. I'd originally assumed that leveling up increased your storage, but it doesn't appear to work that way, since I'm now Level 11 and have unlocked no more storage.

What do I need to do in order to unlock more storage capacity?

As with many things in the game, this is a mechanic that isn't really explained to you. According to the wiki, there is a Maintenance Shop on the Bunker that will sell you chip storage upgrades. An alternative is the female android in the Resistance Camp who sells Pod Programs. She also sells the storage upgrades. Buying from one seller will cause the other seller to no longer have it in stock, though, so you can't double-dip.

The shop appears to sell 2 +8 storage upgrades for 5,000G each, an additional 2 +8 storage upgrades for 10,000G each, then 2 +16 storage upgrades for 20,000G each, and finally, a single +32 storage upgrade for 30,000G.

Combined with the starting storage of 32, you can have a maximum of 128 storage space for chips.