Generally speaking, the dividing line between the Latin alphabet and the Cyrillic alphabet in Eastern Europe is down to the historic/medieval dividing line between Catholic and Orthodox Christianity (in broad strokes). Because of the important role of Latin in Catholic areas, when it came time to develop scripts for the languages in question, they adapted Latin to the task. In Orthodox areas, Glagolitic script was purpose built to act as a script for Slavic languages by St. Cyril. It became the official script of the Orthodox Church in those Slavic areas (in the form of Old Church Slavonic). So, that was how they adopted what would become the Cyrillic script.

The primary exception to this is Romania, which is largely Orthodox. However, since Romanian is a Romance language, it uses the Latin script, rather than Cyrillic. It also bears mentioning that much of what is today Romania was part of the Roman Empire.

The other main exceptions are Bosnia, where both Latin and Cyrillic scripts are used, and Albania, where a Latin script is used. Both countries are historically Muslim and have used a wide variety of scripts. Bosnian once used an Arabic script (alongside Latin and Cyrillic) and Albanian has used Turkish, Latin, Greek, and Arabic scripts. But with standardization in the nineteenth and twentieth centuries both languages ended up adopting Latin alphabets as standard, though I'm not entirely clear on the processes by which this happened.