The reason is purely aesthetic. A watch reading 10:10 has a clean, symmetrical look. Also, the hands don't obscure the watchmaker insignia, which is usually located directly below 12 o'clock. Apparently, the practice started in the 1920s, and it stuck.
10:08, 8 minutes past 10, is roughly the time analog watches are set to in most advertisements, though actual times shown vary between about 10:08 and 10:10. There are several reasons offered by watch companies, many of them psychological, and none of them verifiable as the actual origin of the practice:
The form of the hands has a positive effect on the viewer: the short hand pointing at 10 o'clock and the long hand pointing at 8 minutes is reminiscent of a check mark, which commonly means "ok" or "fine." Some observers further identify this appearance with a smiling face.
The position of the hands does not obscure the date on watches with a date-function at 3 o'clock or any other functions at 9 or 3 o'clock.
The position of the hands does not obscure the company logo, which is often printed under 12 o'clock.
The hands are nearly symmetrically balanced on the face of the dial at 10:08. The minute hand is 48° right of vertical, while the hour hand is 56° left of vertical. Exact symmetry would be achieved at 120/13 minutes past 10:00, approximately 10:09:13.8. Other symmetrical times would not meet the needs above.
At 10 o'clock in the morning, the day is young. There is still time to accomplish many things.
10 o'clock is the time at which people typically wake up if they are able to sleep in. 10:08 is thus associated with weekend, leisure and relaxation.