For this you'll need a tally(numbers) table, that you can create and populate in the following way
CREATE TABLE tally(n INT NOT NULL PRIMARY KEY);
INSERT INTO tally(n)
SELECT a.N + b.N * 10 + 1 n
FROM
(SELECT 0 AS N UNION ALL SELECT 1 UNION ALL SELECT 2 UNION ALL SELECT 3 UNION ALL SELECT 4 UNION ALL SELECT 5 UNION ALL SELECT 6 UNION ALL SELECT 7 UNION ALL SELECT 8 UNION ALL SELECT 9) a
,(SELECT 0 AS N UNION ALL SELECT 1 UNION ALL SELECT 2 UNION ALL SELECT 3 UNION ALL SELECT 4 UNION ALL SELECT 5 UNION ALL SELECT 6 UNION ALL SELECT 7 UNION ALL SELECT 8 UNION ALL SELECT 9) b
ORDER BY n;
Now your query might look like
SELECT name, sdate + INTERVAL n.n - 1 DAY stay_date
FROM table1 t JOIN tally n
ON n.n <= edate - sdate + 1
Output:
+-------+------------+
| name | stay_date |
+-------+------------+
| John | 2013-12-01 |
| John | 2013-12-02 |
| John | 2013-12-03 |
| John | 2013-12-04 |
| Will | 2013-12-04 |
| Will | 2013-12-05 |
| Will | 2013-12-06 |
| Smith | 2013-12-02 |
| Smith | 2013-12-03 |
| Smith | 2013-12-04 |
| Smith | 2013-12-05 |
+-------+------------+
Here is SQLFiddle demo