You don't. These are old platformers that use a password system, whenever you finish a level they give you the password for the next one. Or you can just look it up and enter the passwords for the level you were on when you quit the game.
The windows 95 version had saves and a level select
The Whole Can O' Worms, which is the version GOG sells, is not the Windows 95 version, it's the DOS one. Which is both better and worse. Earthworm Jim games have loads of different versions, each with slightly better and worse things than each other. Some extra levels on some, better sound in others, crispier graphics, manual saves...
But the GOG version (The Whole Can O' Worms), being the DOS one, lacks the save system, you *do* have to use the archaic console password system.