In this case, ' is not a comment character. It's used to delimit string literals. The comic artist is banking on the idea that the school in question has dynamic sql somewhere that looks something like this:
$sql = "INSERT INTO `Students` (FirstName, LastName) VALUES ('" . $fname . "', '" . $lname . "')";
So now the ' character ends the string literal before the programmer was expecting it. Combined with the ; character to end the statement, an attacker can now add whatever sql they want. The -- comment at the end is to make sure any remaining sql in the original statement does not prevent the query from compiling on the server.
FWIW, I also think the comic in question has an important detail wrong: if you're thinking about sanitizing your database inputs, as the comic suggests, you're still doing it wrong. Instead, you should think in terms of quarantining your database inputs, and the correct way to do this is via parameterized queries.