The "gunk" is basically dust and oxidisation built up over the years, natural "rusting" really, that forms a dirt barrier to prevent the game cartridge working as it used to do.
This is a very basic tutorial that costs nothing to fix the problem.
For the purpose of this demonstration, I have used a Nintendo SNES game cartridge however of course this tutorial will work for any other game cartridge as the principles are exactly the same.
Step 1: Open up game cartridge
Easy and cheap method is to get an old screwdriver and file the end into a groove so that it just fits over the recesses in the screw and will therefore turn it.
Then remove the screws.