Hegemony Credits are used to unlock new weapons/items in the Gungeon.
You will eventually find a key to a cell as you wander through the Gungeon; I found mine on level 2. The key will free a shopkeeper, who was also on level 2 for me, though I'm not sure if she'll always be on the same floor as the key, who will then permanently appear in Breach, to the left of the Gungeon entrance.
She will explain things to you there, but basically, you spend hegemony credits to unlock weapons/items that will then have the chance to appear in future Gungeon runs.
Additionally, as stated by agent86, there are other NPCs scattered throughout the gungeon who will offer more unlocks using the hegemony credits!