In this case, "Init" is short for "Initialisation"

Thus, the purpose of the init string is to Initialise the modem; that is, to set up all its features and options according to the requirements of your application.

Common things that you might want to set using an init string are:
Enable or Disable Auto-Answer;
Enable or Disable "Blind" dialling (ie, dialling without waiting for dial tone);
Select Terse or Verbose Response Strings;
Configure any data-compression and/or encryption settings;
etc, etc, etc,...

Obviously, you will need to read the modem's technical manual to determine what options & settings are available, and to find the necessary commands to configure them to your particular requirements...