Modding support is currently available for the Steam and GOG release of the game.
To access mods maps downloaded from the Steam Workshop, head to the mail room (see below image) inside the Spaceship and talk to the mafia located there.
(Please note that 8 Time Pieces and the Ice Hat are required to access the mafia in the mail room)
To access non-map mods (such as cosmetic items like badges, hats, character/weapon skins or dyes), pause the game and click on the "Mods" icon at the bottom.