How does MySQL return a result set from a stored procedure?
Each SELECT statement that does not insert into a table or a variable will produce a result set.

If you want your stored procedure to return only one result set, make sure that you only have one SELECT statement. If you have other SELECT statements, make sure that they insert results into a table or variable.

UPDATE
Here are examples of stored procedures.

This stored procedure would return one result set:

DELIMITER ;;
CREATE DEFINER=CURRENT_USER PROCEDURE stored_procedure_name()
BEGIN
    DECLARE local_variable_name INT;

    SELECT column_name FROM table_1 LIMIT 1 INTO local_variable_name;

    SELECT * FROM table_1;
END;;
DELIMITER ;


This stored procedure would return two result sets:

DELIMITER ;;
CREATE DEFINER=CURRENT_USER PROCEDURE stored_procedure_name()
BEGIN
    DECLARE local_variable_name INT;

    SELECT column_name FROM table_1 LIMIT 1 INTO local_variable_name;

    SELECT * FROM table_1;

    SELECT * FROM table_2;
END;;
DELIMITER ;