Basically it means you can tell the operating system to run the code in the file. For example, if the file was a binary executable, write access would allow you to modify it, read access would allow you to view it, but without execute permissions you would not be able to run the program. In the case of a script, its a little more complicted, because you don't necessarily need to 'run' the program, you can just read its contents into an interpreter, which itself has the execute privelige, but you do not need execute permissions on the script itself.
Some scripts in Linux are themselves executable, you will often see a line at the top like
That line tells the kernel that the file can be executed by calling the relevant program (and isn't just text). Then you can just run your script like
instead of having to do