Sure, these properties control the state of the handshake signals. Their use is not arbitrary, a properly designed serial port device pays attention to them. DTR is Data Terminal Ready, normally connected to DSR (Data Set Ready) on the device. The device assumes that your computer is simply not turned on or the cable is disconnected when DSR is off. It won't send anything and ignores anything you send to it when the signal is off.
RTS is Request To Send, normally connected to CTS (Clear To Send) on the device. Normally used for flow control, it prevents the device from sending too much data and overflow the receive buffer. A nasty problem that is very hard to recover from, the data is entirely lost.
You should normally set the SerialPort.Handshake property to HandShake.RequestToSend so the driver does this automatically. A very common bug is to leave it set to Handshake.None, now you have to turn on these signals yourself. And you'll risk buffer overflow of course, albeit that you'd have to write very slow code to ever get in the danger zone. It has been done.
These signals can be used in hobby projects to control, say, a reed relay. Do beware that the voltages on the signal lines are unpredictable (swings between +/- 5 to 24 Volts) and can't supply a lot of amps (usually 20 milliamps max). You need at least a diode, typically a transistor to switch a heavier load. Ask about it at electronics.stackexchange.com