The salute is an old, old military custom. In the area that I study, the English navy was attempting to enforce a salute to English ships in "English waters" dating back to at least the 1200s, but the custom of saluting another ship, another flag, another port or another admiral became more standardized over time. Originally the salute was meant to show that a ship was disarmed, and ships would fire all their guns (in practice, particularly by the Elizabethan era, this meant that gunners felt free to blaze away at will, and the Ordnance Board complained about the great waste of powder and shot). It's commonly thought that Samuel Pepys was among a group who proposed regularizing salutes by number of guns, and made them an odd number, because even numbers were used for funerary salutes.

In the period I'm more familiar with, roughly the turn of the 19th century, the number of guns a ship would fire (or the number of times fired) would be closely tied to the rank of the person being saluted, with 21 guns reserved for a head of state or a salute to a flag (symbolic of a nation).

Current regulations in the United States provide for 21 guns for a president or other head of state, 19 for a vice president, and decreasing numbers from there.

So why 21 guns at a military funeral? The commonly cited reason is that the dead service member is accorded the same respect as the nation he or she served, and/or that the service entitles them to the same honor as a president.