Let me preface this by saying that we don't exactly know the when.

Why is maybe easier. Note that, across the numeral system of spoken Arabic, numerals are not as simple or straightforward as in English. The system is not readily transparent, and has been the focus of a number of studies on child language acquisition, given that things like case vary so significantly according to the numeral in question.

A bit of history of the written system:

Initially 1865 would not have been written as 1865 (or ١٨٦٥ for that matter), but rather هفضغ. As with Hebrew (א = 1 ,ב = 2, etc), a numeral system existed in Arabic that was based not on separate numerals, but on the letters of the alphabet. So prior to the development of ٥ in the ones place, we would have seen ه, and written (if read left to right) in the same order as "1865" is today.

It is from that system, something like the modern ١٢٣-style numerals developed. A lot of arguments have been made that Arabic numerals are derived from Indic numerals, however Brahmic numerals for which 1 2 and 3 look more like 一 二 and 三 respectively (not Chinese, but I can't type Brahmi on my computer, and they basically look the same) and we have Arabic texts showing a system based on the Arabic script which shows that they did not in fact develop the way Wikipedia (et alia) suggest.

Of course, a lot of scripts have used letters to stand in for numbers (the Romans being an obvious example). What changed was that, rather than writing ف for 60 and و for 6, the system developed to that regardless of the place in the order, a single glyph came to represent 6 at each order of magnitude, and 1865's هفضغ then became ه وح١.

Back to the order. The initial order of reading was right to left. As algebra was being developed and spread through people like Fibonacci, who himself pushed for adopting the Arabic numerals, the order in which the numbers were read would have been shifted following the order of the context language. In an Arabic context, they continued to be read right to left, while in the European context, they were read left to right. Hence the development of different prioritisation of significant digits.

When it changed is a little harder to pin down. In early Qur'anic inscriptions it followed the older order. However within a few centuries, the order was shifting. We know this because within the text, when the verse count was given not in the abjadi numeral system but written out, we can see a shift from "one and twenty" to "twenty and one". We obviously don't have recordings of people speaking from the period, but in cases like this there are what amount to transcribed accounts of how the numerals would have been read out.

It may be impossible to work out precisely when the change would have happened where a majority of speakers were reading the units last. I personally don't know of any written primary historical account that addresses that, though it has been many years since I've been following Semitic linguistics research closely, so it's possible it's there and I've just never encountered it (despite having looked into this exact topic long ago). However, whenever it would have happened, it would have happened quickly. In speech, likely within a generation, even if in written accounts it maintained the old order for a period of time.

We do see early texts, and some as recent as the 15h century, showing an order of the least significant digits similar to how it is in English, where units follow tens. In the late 30s and early 40s there was a lot of discussion on this topic in the Journal of the Royal Asiatic Society. See for example Bailey's Arabic Numerals in 1937, in which he cites a Qur'an which uses orders like خمسون و ثمانية rather than ثمانية و خمسون. A response by Nabia Abbot in 1938, also in JRAS, confirms this as something happening with early Islamic texts but also showing up as late as the 15th century.

To my knowledge, at least in the English-medium literature, there hasn't been a considerable discussion of this in recent decades nor further accounts of attested examples. I would be happy to be proven wrong.

As to why: Much of the Arabic numeral system is learned by rote. There isn't one unified way of handling numerals in Arabic and for each, a learner must essentially memorise (unconsciously in most cases) things like case and how it interacts with what's around it. That's not to say its so opaque as to be unusable; it's not. But it's more than just "2 trees, 3 trees, 11 trees, 12 trees". So, given that the system is inherently one where systematic analysis doesn't buy you much as an L1 learner, that other features of the system would not follow in what one might assume (in this case the reading order of the numeral glyphs) is not inherently problematic, although it is something L2 learners often remark at.

In other words, for a system which is already opaque, additional forms of opacity do not cause conflict. You learn the language you learn as it is used, not the way that an outsider or later analyst may think is more logical or makes more sense. Language develop organically and even when linguistic engineering is imposed, they react to that organically, and this often leads to things that seem illogical. As long as there's no hindrance to communication, or as long as there's some other reason for the system to exist even if it does seem to cause some hinderance (e.g. liturgical languages maintain greater complexity than natural spoken languages, and this in turn can have an influence on spoken language developments as a sort of ideal to which people are encourage to adhere), those things can remain.

So we don't exactly know when the shift would have happened in speech if it happened at all (rather than just being something that showed up from time to time in writing), but it would have been over the course of a couple generations and unlikely to have happened everywhere at once. In writing it existed in both forms up to the 15th century, but was probably a minority of cases where the order was units after tens. We don't have (to my knowledge) a written account of when the change happened, and (to my knowledge) someone has not published the textual analysis to answer that question.

I'm a bit annoyed that nearly all of my Arabic-relevant books are locked up in storage on another hemisphere, or I'd have more to offer. I hope this at least helps a little bit, even if the answer is mostly "we can't quite say".

See also:

• Palmer's A Grammar of the Arabic Language for values and overview of the abjad numeral system (although he continues the story that they are Indic in nature, which I don't believe is tenable at this point in time).

• Forbe's A Grammar of the Arabic Language which also addresses the system but in greater detail

• Boucenna's Origin of the Numerals (actually almost any of his work on the topic) if you want to see some cool old manuscripts that show the earlier forms. The India explanation annoys me for how willingly people just accept it on its face when, logically as well as with documentation, it doesn't really hold up.

(edited to include things from below to save people from nested conversations)