Like you said, the B♭ chord creates a temporary feeling of perhaps being in the key of F. The introduction of the B♭ note triggers a re-thinking or re-consideration or re-calibration of the harmony in your mind, and to many people it is a good feeling. You can do a roughly equivalent thing by playing a Gm chord instead of B♭.
A similar change but towards the key of G is done by playing a D major chord. C - D - G. And then back with the reverse action, G - F - C. The C - D - G movement is so commonly used that the D chord has been given its own name: "secondary dominant". The name is much fancier than the actual thing.
This loops between feelings of C and G majors: C - D - G - F (repeat)
This loops between feelings of C and F majors: C - B♭ - F - G (repeat)
The B♭ chord doesn't necessarily have to lead towards F major. It's just an out-of-scale thing and it sets your thinking process in motion. "What's happening now?"
Move towards Eb: C - B♭ - E♭
Move towards Dm: C - B♭ - A - Dm
Move towards E: C - B♭ - A♭ - F# - E (chain of similar things)
Move towards Gm: C - B♭ - E♭ - D - Gm
Introduce an out-of-scale note and make the listener re-think the harmony. This has many names, "borrowing" and "modal interchange". If the tonic i.e. home note seems to move permanently and not temporarily, then it's a "modulation". It just feels nice to most people. Why? Who knows. Why do things usually fall towards the earth and not away from it? It's a law of nature. For the vast majority of people it is enough to accept that as a law of nature, and learning about gravity adds nothing. Things just fall down, not up. Call that "gravity" if you need that for social acceptance or something. Hey you need to talk proper language so people won't think you're uncivilized, which would jeopardize your status as a respected member of your community. :)
Any 2-year old knows how gravity works and has no problem accepting it. In a sense your question is equivalent to "why do things move towards the floor when I release them from my hand". Or maybe, why is it fun to throw things around? Toy around with music like a toddler plays with toys, and you'll learn how stuff works. :) This StackExchange is explaining gravity to toddlers by reading them scientific articles from Newton and Einstein, when the toddlers should be given toys. Not "modal interchange" mumbo jumbo.
So what I'm saying is, there are two separate things: (1) What happens harmonically = temporary feeling of possible key or mode change. The OP already figured this out through experimentation. Excellent! (2) Why does it feel good? Now THAT cannot be taught by explaining, it just feels good and that's it, it has to be experienced. Any possible "explanation" using lower-level phenomena is going to be useless rubbish with the primary purpose of satisfying a Western logic-information-knowledge valuing person's need to feel that everything is in scientific control. The OP must be given MORE TOYS to learn that there are many similar temporary changes and that they just feel good.