From the Cauchy-Schwartz inequality for the collections $(\sqrt{\alpha}, \sqrt{\beta}, \sqrt{\gamma})$ and $(\sqrt{\alpha x}, \sqrt{\beta y}, \sqrt{\gamma z})$ we obtain:
$$(\alpha\sqrt{x} + \beta\sqrt{y} + \gamma\sqrt{z}) \le \sqrt{(\alpha + \beta + \gamma)(\alpha x + \beta y + \gamma z)},$$
and so
$$\begin{aligned} & a\sqrt{a+b} + b\sqrt{b+c} + c\sqrt{c+a} \le \sqrt{a\sqrt{a+b} + b\sqrt{b+c} + c\sqrt{c+a}} \\ & \le \sqrt{(a+b+c)(a^2+ab+b^2+bc+c^2+ca)} \le \sqrt{2(a^2+b^2+c^2)}. \end{aligned}$$
similarly,
$$\begin{aligned} & b\sqrt{a+b} + c\sqrt{b+c} + a\sqrt{c+a} \le \sqrt{2(a^2+b^2+c^2)}, \\ & c\sqrt{a+b} + a\sqrt{b+c} + b\sqrt{c+a} \le \sqrt{2(ab+bc+ca)}. \end{aligned}$$

Adding all three inequalities, we get the required one.