It's mostly to do with being versatile - so the unit can accept the widest possible range of signals.
An amplifier has what's called it's "dynamic range", which is the range between the smallest signal it can pass (before amplifier noise becomes a problem) and the highest it can pass (when the amplifier starts to clip). It's vital to keep within this range, and preferably within a smaller section of it, where performance will be the best.
It this particular example, you have a filter in the middle, and this will be even more signal level dependent than a plain amplifier. So the first amplifer is to get small signals amplified to a suiatable level to feed the filter - and it has an attenuator in front so it can take larger signals if required.
You can't generally feed a meter-movement directly, so there's an amplifier feeding that, to increase the sensitivity of the meter. The attenuator in front of that is to allow the meter a wider operating range, just as with your multimeter.
Having done a gig with my daughters band last night, perhaps I can use an audio example to try and make it more clear?.
You have an audio mixer for the band, with various vocal mikes and guitars etc. connected to it. EXACTLY like your diagram these first enter an attenuator, which sets the gain of the input (sometimes it actually alters the gain of the stage directly, by adjusting the feedback round an opamp). If you're direct injecting the guitars (or a keyboard), then the signal level is a LOT higher than a microphone, so you need a LOT more attenuation. There's then an amplifier stage that boosts the signal to a good level, well above the noise floor, but well before the onset of clipping. This signal is then fed through tone control stages (the filter stage in your instrument) - after that it goes to another attenuator control (the slider controlling the volume of that stage), it's then amplified again (just like in your instrument) and sent out to the power amplifier to drive the speakers (the meter-movement in your instrument).
This is all done to keep the signal at an optimum level as it passes through the mixer - just like in your instrument!.