Some experts (and non experts) have called LHC the most powerful microscope in the world. I am wondering, what does the LHC magnify? Or is calling it a microscope just wrong and misleading?
Well, it is a microscope if the word is understood in a generalized sense that it allows us to learn about microscopic structure of some stuff. To understand the connection one should first recall how good old optical microscopes work.
The principle is really quite simple, you throw some stuff at the object you are interested in and observe what happens. In the old setting, you throw photons (i.e. light) which then come back to you and you record where they landed (obviously you also need to use some optical methods to actually magnify stuff). This then shows you the structure of the microscopic matter. But at certain point this stops working since visible light has wavelength of roughly $500 \,\rm nm$ so it can't really detect anything smaller than this.
So, to increase resolution, you start throwing smaller things (with smaller wavelength or equivalently bigger energy) at stuff. So you try electrons and neutrons, even photons, but this time with shorter wavelength (e.g. gamma rays). But as you come down to this minute level of atomic and subatomic scales, quantum properties will manifest strongly and the good old approach of throwing photons and seeing what comes back stops working.
At quantum level you can only predict probabilities. E.g. that if you hit some object with a photon, you can calculate the probability that certain particle (e.g. pion) will be created and fly off into the given direction. There is really no way to tell beforehand. Also, at these energies new particles will be abundantely created since you get to energies above the mass of those particles. Therefore at these scales (and LHC is very deeply in these scales), microscope really means collider.
I would add as an answer to "what does the LHC magnify" : it magnifies the microcosm.
Even though it is only probabilities that we can measure, the range is accessible, and the higher the energy, the finer the details of the region explored.