Inductive effect and hyperconjugation are two distinct phenomena.
Inductive effects are purely through sigma bonds. For example, in an organofluorine compound, the fluorine withdraws electrons from carbon by not sharing the electrons of the sigma bond equally. We explain this using the rationale that fluorine is more electronegative than carbon.
Similarly, in a carbocation, the electron-deficient carbon is more electronegative than any substituent carbons. Thus, the carbocation will withdraw electrons from substituents through the sigma bonds.
Hyperconjugation is electron donation of a filled bonding orbital to a nearby low-lying, unfilled orbital. Considering t-butyl cation as an example, a filled C-H bonding orbital can donate electrons to the empty p-orbital on the electron-deficient carbon. From a molecular orbital view, this essentially serves to lower the energy of the C-H sigma orbital and raise the energy of the p-orbital. With only two electrons in this system, the effect is stabilizing.
Notice that this means there are two effects that explain why greater substitution increases carbocation stability: inductive effect and hyperconjugation.
Hyperconjugation can also be invoked for the substitution of alkenes. In this case, the filled C-H sigma orbital overlaps with the empty C-C pi-antibonding orbital.
A good rule of thumb is that a filled orbital overlapping with an unfilled orbital is a stabilizing interaction.