Let $r_1,r_2,r_3$ denote the Cartesian components of $\vec{r}$. All that is needed about spherical coordinates is
$$r = |\vec{r}|= \sqrt{r_1^2+ r_2^2+r_3^2}, \quad \frac{\partial r}{\partial r_j}= \frac{2r_j}{2 \sqrt{r_1^2+ r_2^2+r_3^2}}= \frac{r_j}{r}, \quad \frac{\partial f}{\partial r_j} = \frac{\partial f }{\partial r} \frac{\partial r}{\partial r_j} = \frac{\partial f }{\partial r}\frac{r_j}{r}$$
Using the Levi-Civita symbol, $\varepsilon_{ijk}$ and its property $\varepsilon_{ijk}\varepsilon_{imn} = \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km} $ we have with the Einstein summation convention
$$(\vec{r}\times\nabla)\cdot (\vec{r}\times\nabla)\psi = \varepsilon_{ijk}r_j\frac{\partial}{\partial r_k}\varepsilon_{imn}r_m\frac{\partial \psi}{\partial r_n} = (\delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km} )r_j\frac{\partial}{\partial r_k}\left(r_m\frac{\partial \psi}{\partial r_n} \right) \\ = (\delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km} )r_jr_m\frac{\partial}{\partial r_k}\left(\frac{\partial \psi}{\partial r_n} \right) + (\delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km} )r_j\frac{\partial r_m}{\partial r_k}\frac{\partial \psi}{\partial r_n}\\ = \underbrace{r_j^2\frac{\partial\psi}{\partial r_k^2}}_A- \underbrace{r_jr_k\frac{\partial}{\partial r_k}\left(\frac{\partial \psi}{\partial r_j} \right)}_B+ \underbrace{r_j\frac{\partial r_j}{\partial r_k}\frac{\partial \psi}{\partial r_k}}_C- \underbrace{r_j\frac{\partial r_k}{\partial r_k}\frac{\partial \psi}{\partial r_j}}_D $$
As a reminder, the Einstein summation convention gives $r_j^2 = r_1^2 + r_2^2 + r_3^2 = r^2$. Simplifying each of the four terms on the RHS, we get
$$A= r_j^2\frac{\partial\psi}{\partial r_k^2} = r^2 \nabla^2\psi$$
$$B = r_jr_k\frac{\partial}{\partial r_k}\left(\frac{\partial \psi}{\partial r_j} \right) = r_jr_k\frac{\partial}{\partial r_k}\left(\frac{\partial \psi}{\partial r}\frac{r_j}{r} \right)\\= r_jr_k\frac{\partial}{\partial r_k}\left(\frac{\partial \psi}{\partial r} \right)\frac{r_j}{r} + r_jr_k\frac{\partial \psi}{\partial r}\frac{\partial r_j}{\partial r_k}\frac{1}{r}+ r_j^2r_k\frac{\partial \psi}{\partial r}\frac{-1}{r^2}\frac{\partial r}{\partial r_k}\\ = \frac{r_j^2r_k^2}{r^2}\frac{\partial^2 \psi}{\partial r^2} + \frac{r_j^2}{r}\frac{\partial \psi}{\partial r}- \frac{r_j^2r_k^2}{r^3} \frac{\partial \psi}{\partial r}= r^2 \frac{\partial^2 \psi}{\partial r^2}$$
$$C= r_j\frac{\partial r_j}{\partial r_k}\frac{\partial \psi}{\partial r_k}= r_j\frac{\partial \psi}{\partial r}\frac{r_j}{r} = r \frac{\partial \psi}{\partial r}$$
$$D = r_j\frac{\partial r_k}{\partial r_k}\frac{\partial \psi}{\partial r_j}=3 r_j \frac{\partial \psi}{\partial r} \frac{r_j}{r}= 3r \frac{\partial \psi}{\partial r}$$
Thus,
$$(\vec{r}\times\nabla)\cdot (\vec{r}\times\nabla)\psi = A- B+ C- D = r^2 \nabla^2\psi - r^2 \frac{\partial^2 \psi}{\partial r^2}+ r \frac{\partial \psi}{\partial r}- 3r \frac{\partial \psi}{\partial r} \\ =r^2 \nabla^2\psi - r^2 \frac{\partial^2 \psi}{\partial r^2}- 2r \frac{\partial \psi}{\partial r}$$