This is standard procedure for electric guitars. See, a guitar amp is almost rather part of the instrument than just a loudspeaker: it critically shapes the sound. You might say the speaker cabinet is a substitute for the body of an acoustic guitar: its primary purpose may be to just get you heard, but there's a lot more than just loudness to the art of building well-sounding guitars. You want to boost a wide range of frequencies, but some more and some less†.
So, you get a very different result if you just plug the guitar into the PA, compared to the “proper” amp sound that's usually desired. Of course, if the amp is loud enough to get to all the audience then you don't need a PA at all for guitar – but such a loud amp can be quite a pain for the other performers on stage. It also isn't great for overall band sound because it bleeds in the vocal mics etc.. And for bigger venues, even a couple of Marshall stacks may not be able to get the guitar sound satisfyingly to the back rows. Modern PAs are much more effective at this.
Hence it's usually better to have only a relatively quiet guitar amp (merely loud enough so everybody on stage can hear it enough, particularly the guitarist herself) but use a mic to bring its sound to the audience too, much like you could use a mic to bring the sound of an acoustic guitar to the audience. (Just, because even a quiet amp is much louder than an acoustic guitar, it's much easier to actually get it heard well.)
What's a bit strange about this particular setup is that this seems to be an acoustic-guitar amp. Acoustic amps are different from electric-guitar amps: they are designed to not shape the frequency response much, but give the clearest possible representation of the guitar's own sound. Hence it makes IMO little sense to mic an acoustic-guitar amp. But some guitarist prefer to do it this way nevertheless.See RockinCowboy's comment below.
†Most electric-guitar cabinets are strong in the midrange (with different scoops and peaks), which is good for a powerful presence in the mix, but they cut off the high treble range rather strongly. This is particularly important for the distorted sounds typical of rock/metal guitar; those would be really annoying if you played directly into the PA.