What does "600 : 600 Ω" mean for an audio transformer?
Impedance is just the ratio between voltage and current, like a resistor.

A transformer can change the ratio between in- and output voltage (and current as well) for AC signals.

So a 600 ohms to 4 ohms transformer lowers the voltage (and increases the current) so that 4 ohms at the output behaves as 600 ohms at the input. 

That is useful when you want to connect a 4 ohms speaker to an amplifier which can only handle 600 ohm loads.

A 600 to 600 ohm transformer can indeed be an isolation transformer for an audio distribution system or a telephone line. The transformer is 1 : 1 meaning in- and output voltage stay the same (and the current as well).