Why does the Kremlin in Russia have Arabic-style round towers?

It's kind of the other way around. The building you have in mind is actually St. Basil's Cathedral, which was built in the mid-16th century. However, the use of large towers with rounded domes goes back to at least the 11th century in Eastern Europe, and is probably a development of the types of Church architecture that were popular in the Byzantine Empire in the 10th century when large monumental churches begin to be constructed in what it today Western Russia.

Similarly, Islamic archiecture, especially after 1400 or so, was heavily influenced by both Byzantine and Persian architectural designs. This level of influence is somewhat disguised by the large number of mosques in what is today Turkey and Syria that are were originally Eastern Orthodox churches, but have been heavily modified in the subsequent centuries.

The very fact that you think of Onion domes as "Arabian" is largely a result of a mis-attribution of anything Islamic as also being Arabian somehow. While Islam may have arisen in the land of the Arabian Peninsula, subsquent trends in Islamic Architecture have been very heavily influenced by Persian and Eastern Roman architectural traditions.

This photograph from 1915 shows the original design of the Great Mosque of Kufa, which was the original design of early Islamic mosques. Notice that there is only a series of colonnaded walls, with no roof, and the minaret tower is short and squat. Note that the Great Mosque of Kufa was heavily modified during a series of renovations in the mid-20th century, so the modern mosque on the same site has several smaller towers and additional architectural flourishes that were not part of the original structure.

In contrast, the construction of mosques with large towers really only started in the 1400s, which is why many Persian-style and Ottoman-style mosques have them.