In order to exchange data over a serial connection, the ones and zeroes have to be sent with exact timing, so the receiver can reliably tell where one bit ends and the next begins. Because of this, the hardware that's doing the communication can't do anything else at the same time. And since the actual mechanics of the process are simple and straightforward, most computers with a serial connection have special serial-interface hardware (a Universal Asynchronous Receiver/Transmitter, or UART) to take care of it - the CPU gives the UART some data, then returns to more productive pursuits while the UART works away.
But sometimes you can't use a UART: maybe you're working on a tiny embedded computer without one, or maybe you need to speak a weird 9-bit protocol a standard UART doesn't understand. In that case, you can make the CPU pump the serial line directly. It's inefficient (there's probably more interesting work the CPU could be doing) and it can be difficult to make the CPU pause for exactly the right amount of time (CPUs are normally designed to run as fast or as efficiently as possible, nothing in between), but it's possible and sometimes it's all you've got. That's bit-banging.
But sometimes you can't use a UART: maybe you're working on a tiny embedded computer without one, or maybe you need to speak a weird 9-bit protocol a standard UART doesn't understand. In that case, you can make the CPU pump the serial line directly. It's inefficient (there's probably more interesting work the CPU could be doing) and it can be difficult to make the CPU pause for exactly the right amount of time (CPUs are normally designed to run as fast or as efficiently as possible, nothing in between), but it's possible and sometimes it's all you've got. That's bit-banging.