Let user to add input fields on theme option page

You can do this by adding the following class in your theme for admin options;

Custom-Metaboxes-and-Fields-for-WordPress

And you can call your repeatable fields with some thing like this;

    array(
        'name' => 'Name of field',
        'id'   => 'prefix_option_name',
        'type' => 'text',
        'default'  => '',
        'repeatable' => true,
    ),

For more details on how to use this class you can go through following documentation and example code.