There are many ways to respond to this, a lot of them probably much more useful than this one, but this is how I use mine.
An oscilloscope gives you a visual indication of the difference in voltage flowing at any particular reference point in your gear to whatever choice of reference you want (most likely audio or PSU ground), hence will also show you when there is no signal present, which is pretty essential for testing.
The things to understand from the user manual of your device are the voltage range it's set to (i.e. what range of voltages it will be able to display depending on what range of voltages you are expecting to test), how the trace on the screen is drawn (is it the difference between two signals, the multiplication of two signals, etc. etc. - since an oscilloscope generally provides multiple ways to combine it's inputs), and the scan-rate or for digital the sampling rate of the device - since this defines whether you can see long-term trends (slow sampling rate), or high-frequency spikes, overshoots, or pulses (high sampling rate).
The advantage with a multimeter or oscilloscope for measurement, is that, by their very design, they provide a very high impedance point to add to the circuit you are testing - hence their presence generally goes unnoticed by the circuit you are looking at, and, providing the voltages you are prodding are within the limits of your multimeter or oscilloscope, you can basically put the probes anywhere you want, and get a useful result, provided you are careful not to create a short with the end of the probe.
What a multimeter is good at is reading the steady-state values of DC operating points. What it's NOT good at is precisely what an oscilloscope IS good at - giving you feedback on the AC operation of a circuit. So for audio, an oscilloscope can be used in many ways, but if something is not working, a generally good method is to start at the input, and work your way, with the aid of the schematic, systematically from the input to the output, prodding at anything along the way, and, hopefully even garnering more information about how the circuit works by directly investigating its operation, whilst it's operating.
If you have an unexpected DC offset on the output of a piece of gear, by tracing with an oscilloscope systematically from each point from input to output, you can see WHERE it occurs with an oscilloscope. If you have a signal dropout, you can find the very point that generates the dropout - etc. etc.
As usual this is a pretty open ended question - it depends on what you are testing, what's wrong, and exactly what you want to know, but the real advantage of an oscilloscope is that you can see directly what's going on with your voltages, even if they are RAPIDLY changing (limited by the bandwidth range of your oscilloscope) - something that a multimeter would not provide you to the same level of detail.
If you already know all of this then sorry for the rant
