I'm afraid I can't speak to the legal US format but for both the US letter size and the internationally more widespread DIN (Deutsches Institut für Normung – German Institute for Standardization) "A" format, the sizes of paper are the result of ideas about ratio. Namely, that if you half a sheet of paper you get the next smaller size and if you double it, you get the next larger size.

Two letter size sheets (American National Standards Institute – ANSI A) make one ledger/tabloid sheet (ANSI B) while two A4 sheets make one A3 sheet and so forth. For the DIN A series, the idea is to start with a sheet of 1 m2 and apply a ration of 1 to √2. So with A4 (the most commonly used format), you get 210 by 297 millimetres (8.27 in × 11.7 in). The Letter format applies the same principle with a ratio of 1 : 1.29.

The first writing about standardized paper size based on a folding/doubling ratio is actually quite old because it originated with Georg Christoph Lichtenberg, a German enlightenment writer, physicist, and satirist who lived from 1742-1799. He was the first German professor for what can be referred to as experimental physics in Göttingen. Lichtenberg wrote a letter to one of his friends, detailing the experience of teaching a young man algebra, specifically the idea of a constant ratio. In this endeavor, he noted that already at the time, many printers were using paper based on the 1 to √2 ratio and wondered if this was an accident. He was later going to proof it was not in his book Über Bücherformate where he showed the constant ratio and its meaning in the production process.

The further history of constant ratio paper is strongly tied to the ideas of Enlightenment and its quest for a system of standards and measurements that would enable them to pursue their goals of doing science and spreading rational thought internationally and in a way that enabled everyone to take part in it. In Europe, this resulted in the metric system where based on base units of measurement, 10 is always the number to go by. Like the ratio paper size, it was written about in the Enlightenment and formalized by law in revolutionary France. According to Robin Kinross, one of the seminal authors on the subject of modern typography, the first law to standardize paper came about in 1798 with the law of 13th Brumaire of the Year 7 when revolutionary France instituted paper sheets on the basis of the 1 to √2 for the purposes of taxation and standardization.

While this spread with the Napoleonic wars and subsequent political changes in Europe, it would take until 1917 that the most widespread industrial norm concerning paper size was developed: The DIN A series. As commonly known German was at the time fighting in WWI and in order to aid the war effort, a huge amount of time and resources was invested in standardizing industrial production in order achieve a state in which one factory could easily use the same resources and raw materials used by any other factory in Germany. This also held true for paper sizes – it is simply easier for the industrial process to adapt to a standardized size of things.

DIN's efforts would be mirrored by other countries which also developed Institutes for the standardization of things but due to the fact that DIN was one of the earliest such efforts and were rather efficient in developing them, many others followed the standards it set. Other examples include the standardized transliteration of Arabic into Latin characters. These were adapted by the International Organization for Standardization, founded in 1947 and headquartered in Geneva, which was responsible for efforts to standardize norms internationally.

In 1975, the DIN norm was so widespread that it was formally adapted as ISO 216 standard. By that time, already 88 countries used and it has since then been adapted by all countries in the world except the United States and Canada (albeit in Mexico, Costa Rica, Colombia, Venezuela, Chile, and the Philippines the US letter format is still widespread despite their adaption of the ISO standard).

So, why is the US different and therefore, why are there different paper sizes in the US? The answer to this question is rather simple: The US never introduced the metric system and thus, stayed with the Letter format that works with the 1:1.29 ratio. When the US became independent, they chose to continue with the British system of measurement and despite the gaining popularity of the metric system, simply stayed with it. It became so ingrained in commerce, industrialization, and trade in the US that despite several attempts to change and a 1975 report by the standards bureau that recommended a slow transition to the metric system, it has simply stayed that way.

The exact origins of the original Letter (8 1⁄2 in × 11) size are unknown. According to an old article by the American Forest and Paper Association it was due to the "average maximum stretch of an experienced vatman's arms" in the days of manual paper making, which might be accurate (or not). In contrast, with the metric format, you simply start with a sheet of 1 m2 but seeing as the US eschews using the metric format, that's not really possible for letter size.

So, there it is. The standard letter size might be related to the average length of a Dutch vatman's arm applied to the ratio of 1:1.29 from the days of manual paper making. While the origins are somewhat in the dark, it's continued use is rather arbitrary since when the US standards committee first decided on a standardized paper size in 1921, it decided to use the most widespread format (8 1⁄2 in × 11). For legal paper, having already produced so much paper work by that time, one explanation floating around is that it was kept to maintain backwards compatibility but that is pretty much the realm of speculation and the most likely explanation is rather that it was kept because it had always been this way.

As a kind of follow up, why is this the standard in America while the UK and the (commonwealth) counties I have lived in have the A series paper sizing system and no separate legal sizing as standard? It seems an odd break away for the US to have made.