Oscilloscope to monitor AVR
The bandwidth of a scope is the frequency at which the displayed signal starts to decrease relative to the actual input voltage (at a guess it's the 3dB point where the signal is halved but I've never double checked that).

So if you put a 1V peak to peak signal in at 10Hz the scope will show 1V peak to peak exactly as you'd expect since you are no where near the limit. If you put in the same amplitude at 5.5MHz you'd see 0.5V. If you put in 1V peak to peak at 100MHz (way above the limit) you'd see a constant 0V (the average voltage). 

When looking at signals from the AVR you'd see the correct DC levels and unless they were switching very fast you would see the correct patterns however the corners would show up as being far more rounded than they really are, a 5.5MHz square wave would show up as a sine wave. Even something at 8 MHz would show up to a certain extent but I wouldn't trust the accuracy of any measurements at that point. 

Generally you want the scope to have a lot more bandwidth than the frequency you are working with so that you can see glitches and noise spikes but how vital that is depends on the problems you are trying to track down.

However keep in mind that that scope is an old school analog system, it will be useless for picking up one off transitions or anything that isn't a constant pattern. For digital work on a budget a USB based software oscilloscope running on a PC is the best trade off of price and performance these days. Still no substitute for a good quality standalone scope but a LOT cheaper.