This company keeps calling (man) telling me that I won $250 in free gift cards, but nobody gives away money for free, so it is obviously a scam. I'm worried about how they got my phone number and if my banking information is safe.
You are wise to suspect it's a scam. Your number was probably dialed automatically by a robot dialer. You can stop answering the calls, if you still are, and maybe find a way to block the number. If you don't answer or interact in any way, the calls will eventually stop.