How do bits get transferred over a copper wire?

You won't be surprised to learn it's just varying voltage levels in a circuit formed by the network cables. You also probably won't be surprised to find that the details are fiendishly complicated and far too involved to reproduce here. Even the voltage levels used depend on whether it's 10MHz, 100MHz or GHz cabling. GHz uses five voltage levels and pulse amplitude modulation.

The simplest starting point is probably the Wikipedia article on Ethernet over twisted pair. If you want to know more about how the information is encoded look at the Cisco article on Ethernet.

You might also want to ask on a different SE site e.g. Electrical Engineering or Server Fault.