Do any western countries not use Arabic numerals?
The why is actually pretty simple. It was just a much better system. The hindu-arabic numerical system has two main characteristics that made it superior to for example roman numerals, it is a positional system and it has a constant base (base 10). Positional means the value of a symbol not fixed but it is determined by its position and not just the symbol itself. This is not the case in roman numerals for example (with the minor exception of 4 and 9 and so on) where each symbol has a fixed value and the position of the numbers is mostly a matter of convention and not really functional. This, combined with the use of a constant base meant that one could represent any arbitrary number with a finite set of symbols. More importantly, it made arithmetic calculations much much easier and faster. Just try doing a three digit multiplication or division using Roman or Greek numerals. It would take a while. To understand the advantage of a constant base and how that makes calculations easier, one can just look at the widespread adoption of the metric system compared to the inch-feet-mile combination of the imperial system.
While these properties where not unique on their own, for example the Roman and Greek systems used a constant base (again base 10), and the Babylonian system had some positional elements but varying bases, it was the first such system to combine both in such an effective and elegant way.
It first arrived in the west during the early Renaissance by Italian mathematicians who were introduced to Arabic scholarship. Its adoption by them combined with the fact that they were dominated advances in mathematics in the west during the 15th and 16th century eventually led it its spread in the west.
Eventually, the basic mathematical symbols we know today, such as the equal sign (=) would also be invented and adopted in Italy during that period. Combined with the Arabic numerals and building on the work of Arabic scholars they would transform mathematics from stating problems as "that which when doubled is equal to ten" to an equation "2 x = 10, solve for x", thus creating algebra in its current form and paving the way for advanced mathematics.