How will alkyl radical stability increase?

Hint: Radical stability means the energy level of a radical. Any radical having high internal energy will be unstable, while those having low internal energy will be stable. Radicals are deficient in electrons, and are electronegative in nature. Hence, increasing alkyl radical stability implies the decrease in overall energy of the radical.

Complete answer:
Alkyl groups are substituent groups which are alkanes missing one Hydrogen atom. Some known alkyl radicals are - Methyl radical $( - C{H_3}),$ Ethyl radical $\left( { - {C_2}{H_5}} \right)$ etc. Methyl radical is Methane which has lost a Hydrogen atom while Ethyl radical is Ethane which has lost one Hydrogen atom.
Alkyl radical stability will increase in accordance to these factors: Hyperconjugation and Resonance. Hyperconjugation, also called no-bond resonance, is the delocalization of $\sigma - {\text{electrons}}$ or lone pair of electrons into adjacent $\pi - {\text{orbital}}{\text{.}}$ It delocalized electron density from adjacent Carbon-Hydrogen bonds. As hyperconjugation increases, alkyl radical stability increases. Resonance is the delocalization of electrons over a larger area, meaning charge spread over an increased number of carbon atoms. This effect increases radical stability.
Electron-deficient species are stabilized by neighboring electron-rich species. This implies that increasing the number of alkyl groups in a radical can increase its stability. Therefore, the alkyl radical stability is as follows:
 $
  {\text{Tertiary radical > Secondary radical > Primary radical > Methyl radical}} \\
  {\text{(Most stable) (Least stable)}} \\
 $
Tertiary radicals are those in which a carbon atom is bonded to three other Carbon atoms, while in secondary radicals, it is bonded to two Carbon atoms, and in primary radicals it is bonded to one Carbon atom.

Note:
Alkyl radicals can be stabilized by increasing the carbon chain, as well as being bonded to atoms which have an additional pair of electrons. Atoms like Oxygen, Nitrogen et cetera which are electronegative in nature, when attached to radicals can donate a pair of electrons and increase radical stability.