Why does the Russian alphabet contain letters from the Latin alphabet, but flipped around?

The Russian alphabet does not contain letters from the Latin alphabet. The thing is, both Russian and Latin alphabets got their letters from the Greek alphabet, but in different periods of time. Letters were flipped around in the Ancient and Middle-Aged Greek alphabets, as it happened to many old alphabets. According to the hypothesis, it happened because at the beginning, people had often written their inscriptions on rocks, working from right to left.

Russian alphabet was a Bulgarian alphabet and was originally designed by a Bulgarian monk called Clement of Ohrid. The alphabet underwent a lot of changes and adjustments since then, and it wasn't just a flip

You can read more information in the Wikipedia article on the Cyrillic script (it's pretty informative)

Because it doesn't.

It doesn’t contain flipped letters. The fact they look like flipped Latin alphabet letters is coincidence. Я, for example, looked like ѧ originally, and half of the letter got chopped off over time.