Why does the reaction of the primary alcohol cyclopentylmethanol with hydrogen bromide follow SN1 mechanism?
It does not. The proposed mechanism is wrong.

The conversion of cyclopentylmethanol to 1-bromo-1-methylcyclopentane is not reported anywhere in the literature. If you could somehow generate the primary carbocation, then one would expect a rapid 1,2-hydride shift to generate the more stable tertiary carbocation, giving rise to 1-bromo-1-methylcyclopentane.

The ring-expansion product could be rationalized by a concerted 1,2-alkyl shift displacing $\ce{OH2^+}$ without violating first-principles.