People tell me I shall not use my CB-radio station without an antenna connected or it will result in damage of the device. Is this true and why is this?
You could potentially break some radio transmitters by operating them without the antenna connected.
Several things combine to make this possible.
First, it's difficult to make power at RF frequencies, so the power devices are often fairly fragile, and run near their limiting conditions.
Secondly, radio signals passing down a transmission line behave in a way most non-engineers don't expect. They get reflected from an open circuit, that is, from a connector without the antenna connected.
The antenna provides a load, absorbing the energy travelling down the transmission line from the power device to the antenna. If that energy is reflected, then for a well matched transmitter, it has the potential to double the voltage seen at the transmitting device. Depending on the length of the line, it might alternatively double the current, which usually isn't as bad. So a transistor that's already close to its limits could be pushed beyond them.
If the transmitter's output device is poorly matched, then there's the potential for much higher voltage magnification.
Usually, the transmitter will be designed so that the power device is sufficiently within its ratings that it won't fail into any load, as blowing up into no load is not very nice behaviour. But sometimes, especially for low cost and high power amplifiers, like your CB kicker for instance, it won't be, and needs that load to prevent reflections.
Just to complement the excellent answer of Neil_UK and stress the fact that at RF frequencies voltages and currents don't really behave as those nice entities you know from KCL and KVL.
You must drop Kirchhoff's laws and get your hands dirty with transmission lines theory, where the same concepts of voltage and current become a lot weirder!
In other words, voltages and currents behave more like EM waves that are generated at the transmitter and travel along the cable until they reach the antenna, which "allow them to radiate into space".
Disconnecting the antenna is like putting a brick wall (well, more like a mirror in reality) in front of the wave. The power has to go somewhere, and the open end of the cable cannot make it radiate efficiently, so it bounces back almost entirely, until it reaches the final stage of the transmitter, which is not supposed to sink that power (it should operate as a source).
Hence, as Neil pointed out, if there are no protection devices to dump the reflected wave somewhere, all the reflected power will be dissipated inside the amp, which is not usually designed to do that. POOF! (magic smoke escapes!)
In a sense it's like a rock concert: the band on stage is not deafened to death because the huge loudspeakers are directed toward the people gathered in front of them in a big open space. Put a big wall just before the stage and the band will be blown away by the reflected soundwaves!
What destroys finals is avalanche breakdown, often due to parasitic BJT latching, not reflected power. When the output power amplifier detects the high reflected voltage, it enables the foldback circuit, which reduces output power somewhat proportionally, protecting the finals. Yes, transmitting without an antenna can destroy your finals, but due to the high voltage needed to maintain the high current flowing through the pullup inductor, since it can't flow through the non-existent load. The excessive voltage then causes the FET to go into avalanche.