This defines what shell (command interpreter) you are using for interpreting/running your script. Each shell is slightly different in the way it interacts with the user and executes scripts (programs).
When you type in a command at the Unix prompt, you are interacting with the shell.
E.g., #!/bin/csh refers to the C-shell, /bin/tcsh the t-shell, /bin/bash the bash shell, etc.
You can tell which interactive shell you are using the
echo $SHELL
command, or alternatively
env | grep -i shell
You can change your command shell with the chsh command.
Each has a slightly different command set and way of assigning variables and its own set of programming constructs. For instance the if-else statement with bash looks different that the one in the C-shell.
This page might be of interest as it "translates" between bash and tcsh commands/syntax.
Using the directive in the shell script allows you to run programs using a different shell. For instance I use the tcsh shell interactively, but often run bash scripts using /bin/bash in the script file.
Aside:
This concept extends to other scripts too. For instance if you program in Python you'd put
#!/usr/bin/python
at the top of your Python program