I don't understand how the 'cyclopropenium' ion is aromatic. According to my understanding, the carbons within the central 3 carbon ring must each be sp2 hybridised with a single electron in a p orbital - since each carbon is bonded to 3 other atoms and has 4 valence electrons. How does that lead to aromaticity, since this doesn't fullfil Huckel's rule?
It does meet Huckel's rule. When you have $m$ carbon atoms in a conjugated ring with a positive charge of $q$, there are $m-q$ pi electrons. Here $m=3, q=1$ therefore the number of pi electrons is $m-q=2=4×0+2$.
A Huckel model calculation indicates that cyclopropenyl cation has roughly as much stabilization as benzene, but it's distributed over three instead of six carbon atoms. Thus aromaticity is especially powerful in a cyclopropenyl cation ring. This explains how a positive (formal) charge can be stabilized on as few as three carbon atoms and the ring holds together despite steric strain.
the carbons within the central 3 carbon ring must each be sp2 hybridised with a single electron in a p orbital
This is incorrect.
The requirements for Huckel aromaticity are:
There must be a continuous, planer ring of overlapping $p$ orbitals.
There must be $4n+2$ $\pi$ electrons in the system.
The cyclopropenyl fulfills these requirements because it is planar, has overlapping $p$ orbitals, and has 2 $\pi$ electrons ($n = 0$).
The cyclopropenyl cation is aromatic because it is meeting all the definitions of Huckel's rule of aromaticity:
- All carbons are sp2 hybridised.
- There are 3 carbon atoms which form a conjugated system and moreover it has a positive charge therefore it has $3$-$1$=$2$ pi electrons.
- If we put $n=0$ in Huckel's expression ($4n+2$) pi electrons we are getting 2 as the result.
- The cation also has overlapping planar and parallel p orbitals. You may be knowing that for maximum overlapping, orbitals should be parallel to one another and planar.
Since the required criteria are fulfilled, it is aromatic.
It's because the of the small size of the molecule. The hydrogen on the cation does not interfere with pi electrons orbiting around the cyclic molecule.