In the United States, we do it because the United States Postal Service requires that we do.
Why do they specify the upper-right? As near as I can tell, it appears to be based on tradition. The first postage stamps were issued in the United States in 1847. The system before that involved the local postmaster hand-writing the postage on an envelope. The postage could be paid at time of sending or by the recipient. In any case, the postmaster wrote the postage in the upper-right. Later, this was standardized when automatic cancelling machines were introduced in the late 19th century. Think of how when you get a letter, the stamp has been cancelled with an ink stamp. For a while now that's been done automatically, rather than by hand.
The UK introduced the world's first postage stamps in 1840. From the outset, the Royal Mail asked people to put stamps on the top-right of the envelope. This site has an image of a complete sheet of the "Penny Black" (the first postage stamp ever issued, 1840) as well as a complete sheet of the "Penny Red" (1841). In both cases, note the inscription along the margins of the sheet: "Place the Labels ABOVE the Address and towards the RIGHT HAND SIDE of the Letter."
The prescription of a particular location for the stamp on letters was convenient to post office employees, who at that time were required to cancel each stamp by hand. I have however never heard of a stamp not being accepted due solely to its position on the envelope (in the UK or any other country), so it seems like the admonition to put the stamp in the top-right was always a recommendation rather than a commandment.
The UK also introduced widespread, and very successful, reforms of its public postal system in 1840, and in subsequent decades other countries viewed the UK as a model to emulate, both in reforming their own postal systems and introducing postage stamps. I don't know why any other country chose to follow the UK's system of stamp placement (and indeed many seem not to have, at least initially) but as the UK was seen as an example of good postal practices in general, I would assume other countries which followed the UK's lead on stamp placement did so for that reason.
Now, none of this actually answers why the UK chose the top-right in the first place, and unfortunately I can only speculate as to that, so I won't. But at any rate the top-right corner was selected as the optimum location for the first postage stamps in the world, and other countries seem to followed the British example.