How does one increase their arm strength without weights?

Without a weight of some kind you cannot.
Even holding a rock at arms length until you arm does not waver.

If you don't have weights or resistance bands, then pushups and pullups will help with arm strength.

Pushups, pullups, dips (using a chair or two chairs)...All pretty much standard. More advanced stuff includes the many variations of pushups, "one arm" pull-ups (holding the wrist with one arm) "muscle ups" (as done by gymnasts) handstand presses....
There are all sorts of variations of these...Google "bodyweight exercises".