What is hyperconjugation?

Hyperconjugation is the overlap of a ##σ## bond with an adjacent empty or partially filled ##p## or ##π## orbital to give a delocalized molecular orbital.
Only electrons in bonds that are ##β## to the positively charged carbon can take part in in hyperconjugation.
This overlap of orbitals and delocalization of electrons increases the stability of the system
Hyperconjugation explains the relative stability of carbocations:
(CH₃)₃C⁺ > (CH₃)₂CH⁺ > (CH₃)CH₂⁺ > CH₃⁺
The C–C ##σ## bond adjacent to the cationic carbon is free to rotate.
As it does so, the three C–H ##σ## bonds of the methyl group in turn overlap with the vacant ##p## orbital.
The more adjacent C-H bonds there are, the larger the hyperconjugation stabilization is.
An ethyl cation has three C-H ##σ## bonds; an isopropyl cation has six; and a t-butyl cation has nine C-H ##σ## bonds.
This agrees with the observed order of stability.
Here’s a video that explains hyperconjugation.