How add built-in textarea in theme development?

This is called wp editor, which uses tinymce editor and odds. This editor can be called using wp_editor function.

To display it on your options page, use wp_editor(); Like –

$field_value = get_option('a_custom_textarea');
$field_name="a_custom_textarea";

wp_editor( $field_value, $field_name );

Ref – https://codex.wordpress.org/Function_Reference/wp_editor