how to register / add multiple options into one field in wordpress?

Just set the name attribute in your checkbox tag in the format like: favourite_colors[]. The following code is derived from this thread.

add_settings_field(  
    'favourite-colors',  
    'Select your favourite colour',  
    'favourite_colors_checkbox_callback',  
    'my-settings'
);

function favourite_colors_checkbox_callback() {

    $options = get_option( 'favourite_colors' );

    $html="<input type="checkbox" id="red" name="favourite_colors[red]" value="1"" . checked( 1, $options['red'], false ) . '/>';
    $html .= '<label for="red">Red</label><br />';
    $html="<input type="checkbox" id="yellow" name="favourite_colors[yellow]" value="1"" . checked( 1, $options['yellow'], false ) . '/>';
    $html .= '<label for="yellow">Yellow</label>';

    echo $html;

}