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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.