Contact. Simple as that.

Initially languages were written in even more diverse ways. In Greek, you had texts which used a system known as boustrophedon which would be left to right for one line, then right to left for the next, then switching again. Today, some Greek letters are now standardised as the LTR form, and some as the RTL.

In Classical Chinese, you would write top to bottom. In Japan and Taiwan, many books are still written this way. Here's an example from my own bookshelf, a modern text (Peter Hessler's Oracle Bones). Because of the influence on Korean and Japanese by Classical Chinese and Chinese culture more generally, this same system was adopted.

However, around the start of the 20th century, a lot of countries were looking to modernise, as they saw it. For many this included language/script reform. A big part of this also means changing the direction they wrote (e.g. China), if not the writing system entirely (e.g. Vietnam). Many Asian nations switched their direction to LRT following the European standards. The 20th century also saw the creation of a number of standard orthographies for minority languages. This was a major endeavour for the Chinese government at the time, which as a way of standardising writing for languages like Zhuang, developed Cyrillic and then later Latin based orthographies (as feelings toward the USSR cooled).

Script choices are political, however. In 1918 it was proposed that Uyghur be written in a Latin script, which mostly happened. In the 1950s a new version was developed (as part of the above discission on standards) called Uyghur Yëngi Yëziqi to replace the Cyrillic version (Uyghur Siril Yëziqi) which replaced the former Latin version after 1949. Then in the 1980s an Arabic script was adopted as the standard, known as Uyghur Ereb Yëziqi. Then more recently was the development of a new Latin version, Uyghur Latin Yëziqi, which exists but isn't widely used. It's more just a supplement to UEY. It's unlikely that UEY will be replaced any time soon, since despite Uyghur not being related to Arabic at all, the significance of the Islamic identity is such (as a contrast to Han identity) that it's unlikely to go anywhere unless forced.

Anyway, back to the point: The reason that we now have a multitude of scripts going from left to right is basically for the same reason that we have a lot of countries learning English in the last 50 years. Having a LTR writing system has historically been seen as a sign of being modern. It facilitates use of your language on computers, and if it's Latin based, all the better, because you now have an easy time finding keyboards you can easily use.

Thus, ultimately, the reason comes down to contact.