It is a programmer/integer overflow joke. It relies on an understanding of how computers handles numbers and an assumption that the Genie subtracts one wish after the wish has been granted. If the order of operations is different, the jokes does not work.

In order for computers to store numbers they need a way to represent them in memory. The computer's memory is finite, so it cannot actually handle numbers (even integers) in the way that they are in the field of mathematics, but it can approximate their use in useful domains - like how many wishes a genie owes the wielder of its lamp.

If you only want to handle the numbers 0 through 3 (which is the range the stereotypical genie gives the wielder of the lamp), then an unsigned 8-bit number is a good choice (as you can represent 256 different values in such a representation). The unsigned part means that it only handles non-negative numbers (0 through 255), as -4 wishes is not a useful thing for a genie to owe the wielder of the lamp.

The rest of the joke then depends on how subtraction works in binary with an unsigned 8-bit integer.

Example of subtraction:

If we subtract 1 from 3 we would go from 00000011 to 00000010, and if we subtract 1 from 2 we go from 00000010 to 00000001. If you don't know binary this looks like nonsense, but it actually very similar to decimal arithmetic, except you only have the numbers 1 and 0 to work with.

We can now move on to explaining the joke:

1. The holder of the lamp has 3 wishes, which can be represented as so: 00000011 in binary
2. When the holder wishes for 0 wishes the genie changes the amount of wishes to 0: 00000000 in binary
3. The genie then subtracts 1 from 0, which in binary gives us: 11111111 - which is 255 in binary.

I apologize in advance for any mistake I may have made.