Why it is only carbons that have hydrogens attached (and not just hydrogens directly attached) to the charged carbon which result in hyperconjugation?
Hyperconjugation happens because of an overlap of orbitals.

Consider this: in (uncharged) 2-methylpropane, the central carbon has sp3 hybridisation (and a hydrogen bound to it).

When the carbocation of the image is formed, the hydrogen leaves, the hybridization becomes sp2 (hence, planar), and now there is a free pi orbital available.

Here is what causes hyperconjugation: pi orbitals are quite larger than how they are drawn in books, and they can overlap partially with the orbitals of the adjacent hydrogens, and get part of their charge.

In the example of butane, the thing is the same: when the carbocation is formed, the carbon and the two hydrogens attached become planar (sp2), and a perpendicular pi orbital is formed. It cannot interact with the hydrogens attached to the charged carbon because they are on its nodal plane. Instead, it overlaps partially with the orbitals of the hydrogens on the adjacent carbon.