Remember that all the carbons in [16]annulene have hydrogens attached.
Here's how I would explain the sittuation with [16]annulene. In [16]annulene, every carbon is bonded to two carbons and a hydrogen. Each carbon also participates in a double bond with a neighboring carbon. Together, these say that each carbon is sp2 hybridized and has one electron in the unhybridized p-orbital.
So far, [16]annulene fulfils the first two criteria for aromatic molecules. It has a ring of sp2 hybridized carbons that are coplanar. But, since each carbon contributes one electron to the conjugated pi system, the conjugated pi system has 16 electrons, which does not fulfil the 4n+2 pi electron criteria (14 or 18 would, however).
But, what happens if we take away two electrons? The two carbons would lose the electron in their p-orbitals, generating two carbocations. As usual, the carbocations are sp2 hybridized and have an empty p-orbital. So, in your [16]annulene cation, you still have a coplanar ring of sp2 hybridized atoms. Furthermore, the conjugated pi system now has 14 electrons which fulfills the third criterion for aromaticity.