This may be a slight misattribution. It is the implication $(1.22) \implies (1.21)$ which is essentially in Stein's paper; the implication $(1.21) \implies (1.9)$ is much simpler, following from Marcinkiewicz interpolation. Roughly speaking, if one has a weak-type $L^p$ estimate on the Kakeya maximal operator, one can interpolate it with the $L^2$ estimate (1.20) to get a strong-type $L^{p-\varepsilon}$ estimate, and then interpolate again with the trivial $L^\infty$ estimate to get back a strong $L^p$ estimate, losing some factors of $\delta^{-O(\varepsilon)}$ in the process.
[A more general principle here is that, in situations where one is willing to concede epsilon losses in the exponents, strong type and weak type estimates are equivalent; also, singular integrals are all basically bounded, rendering a large part of Calderon-Zygmund theory trivial in this regime.]