Mass Spectrometry (proteomics): How isotopes can be used to to determine charge of the peaks?
If you have an ion of mass 100 and charge 2, the m/z ratio (let's exclude the root) will be 100/2=50. If you have isotopes, then you expect to find several peaks for a given fragment. For example, the masses of different isotopes of the same molecule could be like: 100, 102, 105 and 107. In this case the spectrometer will detect four peaks, instead of one, for the same molecule (will look something like this): 100/2, 102/2, 105/2 and 107/2. The charge of all that peaks is the same.. it's only the mass that differs.

The page you linked simply makes an example, "singly charged ion will show isotopic peaks that differ by 1" example m=10 z=1 m/z =10, for the isotope m=11 z=1 m/z=11 (difference of 1). If you have a doubly charged ion m=10 z=2 m/z=5 and for the isotope will be 11/2=5.5 (difference of 0.5).

So, if you take the spectra of a compound and you see the isotopic peaks having a distance of 0.5 among each other, you are looking at a doubly charged ion, if the distance is 1, it's a singly charged ion.

What do the isotopes have to do with the charge?
Given that the isotopes peaks are around 1 units far from each other, you can conclude that the ion you are looking at has 1 charge.

The last question is not clear to me.