In Classical Latin, the letter C represented a single phoneme
/k/. How exactly this phoneme was pronounced is still debated—there's some evidence that the pronunciation varied a bit depending on the vowel after it—but loans into other languages suggest that it always remained pretty close to the sound in English "cat". So when people use reconstructed Classical pronunciation, they generally pronounce it like that, regardless of the environment.
In Vulgar Latin/early Romance, the pronunciation shifted more dramatically: now, C before I, E, and a few other letters ("front vowels") was pronounced like the sound in English "chat". This is what we still see in modern Italian. And in the variety of Romance that eventually became French, it shifted even further, becoming the sound in English "sip" (consider c'est, pronounced roughly like English "say").
Now, English is generally considered a Germanic language: it descends from the Anglo-Saxon varieties of Germanic spoken in Britain, not from any sort of Vulgar Latin. But around the 11th century, French-speaking Normans invaded and conquered England, and their language had an enormous impact on the development of English.
So most of the Latinate/Romance vocabulary in English came to us through some variety of French, and was thus affected by French sound changes. Even when words were borrowed directly from Latin ("learned borrowings"), we tend to use a vaguely French-esque pronunciation for them (compare "Caesar" to German "Kaiser"); sometimes these learned borrowings actually came via French, but other times it's just a generalization of the idea "this is how we pronounce Latin words now".