The original question was WHY, and I think we must credit American propaganda with that.
In its quest for a new rifle to replace the already-obsolescent Krag, the US engineered a replacement in 1901 which was, essentially, a copy of the very latest Mauser, the 98. These were made in small numbers and removed when the final version, the 1903, was introduced. The 1903 essentially was a 98 although with a Krag firing-pin assembly and this rifle satisfied the US military requirements. The few 1901s were surplused quickly, showing up 40 years later in .303 calibre as the Bannerman Springfields, 100 of which were given to Great Britain in the desperate period of early WW2.
A whacking huge lawsuit ensued, being that Mauser/DWM were not happy with having their latest and best design virtually pirated from them. In the end, the US paid $200,000 for the patent rights with which to produce their 1903s legally. The world economy at that time recognised the US dollar as 1/20 of a Troy ounce of fine gold. The US therefore paid 10,000 ounces of fine gold for the rights to produce their rifle, which was based solidly on the 98 Mauser. Considering that today's price of gold is $1162.06US to the Troy ounce fine, this would mean a payment of $11,610,600 for the licence to use the Mauser 98 patents. This is not small change, nor is it to be ignored.
Paying THIS kind of money for a design..... and then modifying to US military standards..... the 1903 HAD to be the best rifle in the world. Politically, nothing else was possible.
Actually, the 1903 was in ways a flawed design; it earned its laurels as a range rifle (and it DID earn them, no mistake) although the very high standards of maintenance and tuning must have had at least a bit to do with that. The old practice of swapping barrels every 3000 rounds would have helped here although made partly necessary by the Pyro powder. In contrast, a Lee-Enfield barrel using Cordite Mark I still was in its prime at 3000.... and this improved a good bit with the introduction of Cordite MDT 5-2 in 1910.
The 1903 was never what one could call CHEAP or COMMON. It was a highly-tuned specialists' rifle which only really BECAME a good combat rifle with the introduction of the A3 version.... well after its replacement, the M-1, was in production. Supplies of 1903 rifles were supplemented 2 to 1 during the Great War by the "vastly-inferior British-designed" Model 1917 which likely was one of the BEST WW1 rifles actually to take the field.
But everybody KNEW that the 1903 was based strongly on the 98 so, when large supplies of surplus 98s hit the market, their prices were buoyed by the 1903 legend.... and so, almost a century later, the 98 still is considered to be the acme of perfection.
No doubt the 98 can be a wonderful rifle (NOBODY gets their hands on my 98AZ without my myopic eagle eye on them) but, if you want to take all the propaganda, nationalism and emotion out of the equation, there HAVE BEEN superior designs. As an example, I would urge you to look at a BUFFALO NEWTON (IF you can find one): a 98 ACTION with a Mark III ROSS BOLTHEAD.
One reason for the ubiquity of the 98 is simple: it was EASY TO PRODUCE, once you had the equipment. And good German equipment, used for producing the 98, was scattered about the globe in 1919/20 as the German State Arsenals of Danzig, Amberg, Spandau and Erfurt..... and the massive DWM plant in Berlin..... were stripped of their tooling..... which ended up in Belgium, Poland, Czechoslovakia and assorted other places around the map. 
So there are the reasons: design, national pride, MONEY and the results of a war which nobody in their right mind wanted.
That's how I see it, anyway. Please don't start the flaming for a few minutes: this old man needs a few minutes, cane firmly in hand, to climb into his nice Nomex coveralls!