The utility of the other signals becomes apparent when you think about RS-232 as being the primary way for networking computers, and mostly over modems. Unlike today's common networking physical protocol, Ethernet, the data are not framed, so any out-of-band communication must happen on another wire, with simple electrical interfaces that could be sent and received by the simple electrical interfaces of the day.
Many of the signals come in pairs which differ only in direction.
RXD (receive data) / TXD (transmit data)
RTS (request to send) / CTS (clear to send), DTR (data terminal ready) / DSR (data set ready)
Hardware flow control, to prevent buffer overflows. These each implement a way to assert "I have something to send (RTS)" and for the other end to answer with "I am ready to receive it (CTS)", with DTR/DSR doing the same thing in the other direction. See What is the difference between DTR/DSR and RTS/CTS flow control? on Stack Overflow.
Or, just abused for whatever purpose deemed convenient. Interoperability wasn't super, back in the day. Everyone had a box with LEDs to indicate which lines were asserted so they could figure out what their hardware was doing, and another box with movable jumpers or switches so they could switch the lines around to get two things to talk.
These days, a lot of equipment is fast enough that there is no chance of a buffer overflow occurring, so flow control may not be necessary.
CD (carrier detect)
CD is essentially the DTE (modem) asserting that whatever it's connected to (phone line) is physically connected, off hook, ready to go, etc. Or again, just abused for whatever was convenient.
RI (ring indicator)
Asserted when the phone line rings.