I am assuming that by "scientific names" you are referring to binomial nomenclature, the use of Latin for biological taxonomy (Homo sapiens, Canis lupus, Helianthus annuus, and so on). If that isn't what you're talking about, please feel free to elaborate further.

One could write a lot on how Latin became established as the "language of science" in the 13th century European context, and remained as such until the late 18th/early 19th century. That's a longer story and not a straightforward one (it is not as simple as saying that it "did not die" with the Roman Empire — it had to do with ideas of what Renaissance scholarship meant, and the fact that many recovered Ancient and Roman texts were Latin translations of Arabic translations created by the Byzantine Empire). For this, see Michael Gordin, Scientific Babel, chapter 1. (Or ask another question specifically about this.)

As for binomial nomenclature, this was developed by the 18th-century Swedish botanist Carl von Linné (better known by his Latinized name, Carl Linnaeus). The idea, not entirely unique to Linnaeus but developed and popularized by him, was that one needed a "neutral" language to make a scientific study of plants and animals. Even within a country, there could be separate names for the same things. Certainly between two languages, there are different names for the same species. Linnaeus and other naturalists sought a system that could not only be very flexible and adaptable to the wide world of natural history (and fit well with Linnaeus' taxonomical system in general), but also could be relatively easily used in everyday scientific usage.

OK, so why Latin, specifically? Because Linnaeus only wrote in two languages: Swedish and Latin. He interestingly could not read French, which is what most Swedish elites would have been using when communicating internationally. Given that Swedish was certainly not the language that was going to bring together the sprawling scientific world, or lead to the adoption of a new system of taxonomy, Latin was a much better pick.

One of Linnaeus' students, Torbern Bergman, did further popularizing and expanding of this system, notably applying it to chemistry. It is of note that the Bergman initially tried to use Swedish for this work, and it utterly failed — nobody else wanted to use it, and grammatically it was apparently difficult to make straightforward elaborations on the terms. In one of his final works, he explained his preference for Latin in the end:

In establishing entirely new names, I desire that their origins be Latin. This language is, or at least was, the vernacular of the learned: now it is dead and is not subject to constant changes. Therefore, if the reform is conducted in this language first, and afterward in living languages on the same model (as much as the spirit of each will allow), this same reform will be conducted more easily. For this very reason chemical language can attain general agreement in all places, which promises no small benefit not only in reading foreign works, but also in translating.

The irony here is that Latin was actually on it way out as a general language of science at nearly the same moment it was adopted for binomial nomenclature. By the 19th century, the annoyances of reading and writing in Latin lead to a splintering of European scientific writing into the "big three" (French, German, English), which, by the late-20th century, has largely collapsed to one (English). So binomial nomenclature is actually a curious case: scholars would soon stop publishing their works in Latin, but for the purposes of taxonomy it remained popular.

This comes from the end of the aforementioned chapter by Gordin, where he describes Linnaeus and Bergman's work in some detail.