Admittedly, I haven't done much research on this yet and it's a question that's only recently come to me but what's the point of a AWG?
Is it a calibration thing for your scope? So you know what it's supposed to look like vs what it actually looks like?
As already mentioned, an AWG can generate arbitrary waveforms, but they can also generate all of the common waveforms - sine, square, triangle, DC, etc. These are the signals that you'll use most of the time. In addition, many allow you to generate swept signals (i.e. where the frequency (or amplitude) varies up or down) or maybe modulated signals (AM, FM, etc).
This means that they are ideal for testing of circuits during development, when specific waveforms are required. For example, I'm currently playing around with LED strobe lights, but I'm first interested in seeing how the LEDs behave when strobed. Rather than make up, say, a rather complex 555 based circuit to drive them, I can use a sig. gen. to generate a square wave whose duty cycle and frequency I can vary by turning a knob, at will.
Or maybe you want to study demodulation techniques, then it's nice to have a device that can generate a known "good" modulated signal whose baseband and carrier frequencies you can vary as required.
More generally, I find them useful to "break in" to a non-working circuit - got element A that should generate a sawtooth wave to drive element B but something isn't right? - then disconnect A from B, and drive B with a sawtooth from the sig. gen. This may give you a clue as to what isn't working.
These days these devices can be pretty inexpensive too - I have a cheap'n'cheerful FY6800 that can do all of the above, and it cost about £80, which to my mind is incredible value for money.