First, the critical dimension. There are many ways (seemingly inequivalent ways but ultimately bound to give the same result) to calculate $D=10$ for the superstring that mirror the methods to calculate $D=26$ for the bosonic string.
For the bosonic string, one may use a conformally invariant world sheet theory. Because of the residual conformal symmetry, it has to have $bc$ ghosts. The central charge of the $bc$ system is $c=1-3k^2$ where $k=2J-1$ where $J$ is the dimension of the $b$ antighost, in this case $J=2$. You see that my formulae imply $k=3$ and $c=1-27=-26$ so one has to add 26 bosons, i.e. 26 dimensions of spacetime, to get $c=0$ in total.
Now, for the superstring, the local symmetries on the world sheet are enhanced from the ordinary conformal group to the $N=1$ superconformal group. One needs to add the $\beta\gamma$ (bosonic) ghosts for the new (fermionic) generators. Their dimension is $J=3/2$, different from $J=2$ of $bc$ by $1/2$, as usual for the spin difference of things related by supersymmetry. You see that $k=2J-1=2$ and $3k^2-1=12-1=11$. Now, the central charge of $\beta\gamma$ is $3k^2-1$ and not $1-3k^2$, the sign is the opposite one, because they are bosons.
So the $bc$ and $\beta\gamma$ have $c=-26+11=-15$. This minus fifteen must be compensated by 10 bosonic fields and 10 fermionic fields (whose $c=1/2$ per dimension: note that a fermion is half a boson) and $10+10/2=15$ so that the total $c=0$. If some of the steps aren't understandable above, it's almost certainly because the reader isn't familiar with basics of conformal field theory and it is not possible to explain conformal field theory without conformal field theory. It's a whole subject, not something that should be written as one answer on this server.
In this formalism with the new world sheet fermions $\psi^\mu$ transforming as spacetime vectors, one has to protect the spin-statistics relationship. Vector-like fermions violate it so they are only allowed in pairs. This is achieved by the GSO projection – well, there are actually two GSO projections, one separate for left-movers and one for right-movers. Only 1/4 of the states are kept in the spectrum. The projection is a flip side of having four sectors – the left-moving and right-moving fermions may independently be periodic or antiperiodic. I wrote about the GSO projection a month ago:
Again, if anything is incomprehensible and incomplete, it's because it's not really one isolated insight that a layman may understand from one sentence. It's one of many technical results that follows from a large subject – string theory – that has to be systematically studied if one wants to understand it.