Well, as you can see in the contributed example to the WordPress Reference, it is not that add_settings_field magically knows about a setting being true or false, and it is not limited to checkboxes (boolean flags):
Instead, the function signature states clearly that the $callback parameter it is required. And the introduction says:
The $callback argument should be the name of a function that echoes out the html input tags for this setting field. Use get_option() to retrieve existing values to show.
Maybe it should say that get_option it’s the recommended (as it is the easier) way to show stored setting in the database, but you decide how do you use it in your callback function, and how that control renders the option.
In this example provided, yes, it is a checkbox that is checked (or not), depending on the value stored:
function callback_input_myid() {
echo "<input type="checkbox" id='mynewcheckboxID' value="1""
if ( get_option('mynewcheckboxID') == '1' ) {
echo ' checked';
}
echo '/>';
}
But as I’ve said before, it is not limited to boolean settings and checkboxes therefore. You could store a text, a color and your function render a ColorPicker, to store a date therefore render a DatePicker, even store a file in base64, however I’m not sure that should be advisable, just saying what is rendered and what option is retrieved it is entirely your responsibility.
Put it simple, get_option is just a convenient and pluggable way to do a SELECT FROM options table.