7.35 is nothing more than a definition of spherical coordinates.

⎧    x = r sinθ cosϕ
⎪
⎨    y = r sinθ sinϕ
⎪
⎩    z = r cosθ


7.47 is the set of algebraic conditions expressed by the vector definition 𝐋 = 𝐫 × 𝐩.

⎧    L̂𝓍 = yp𝓏 - zp𝓎 = -ιħ (y ∂͟_  - z ∂͟_ )
⎪                           ∂y      ∂y
⎪
⎨    L̂𝓎 = zp𝓍 - xp𝓏 = -ιħ (z ∂͟_  - x ∂͟_ )
⎪                           ∂y      ∂y
⎪
⎪    L̂𝓏 = xp𝓎 - yp𝓍 = -ιħ (x ∂͟_  - y ∂͟_ )
⎩                           ∂y      ∂y

Substituing 7.35 into 7.47,

⎧    L̂𝓍 = -ιħ (r sinθ sinϕ ∂͟_  - r cosθ ∂͟_ )
⎪                          ∂y           ∂y
⎪
⎨    L̂𝓎 = -ιħ (r cosθ ∂͟_  - r sinθ cosϕ ∂͟_ )
⎪                     ∂y                ∂y
⎪
⎪    L̂𝓏 = -ιħ (r sinθ cosϕ ∂͟_ - r sinθ sinϕ ∂͟_ )
⎩                          ∂y               ∂y


⎧    L̂𝓍 = -ιħ (r sinθ sinϕ ∂͟_  - r cosθ ∂͟_ )
⎪                          ∂y           ∂y
⎪
⎨    L̂𝓎 = ιħ (r cosθ ∂͟_ + r sinθ cosϕ ∂͟_ )
⎪                    ∂y               ∂y
⎪
⎪    L̂𝓏 = -ιħ (r sinθ cosϕ ∂͟_ - r sinθ sinϕ ∂͟_ )
⎩                          ∂y               ∂y