The inequality is equivalent to
$2 a^{3} + 2 b^{3} + 2 c^{3} + 2 a b c \geq a^{2}(b+c) + b^{2}(c+a) + c^{2}(a+b) + 2 a b c$,
hence
$$2 a^{3} + 2 b^{3} + 2 c^{3} \geq a b(a+b) + b c(b+c) + c a(c+a) \tag{1}$$
For any positive real numbers $x$ and $y$ we have
$$x^{3} + y^{3} \geq x y(x+y)$$
Indeed, this inequality is equivalent to $x^{2} - x y + y^{2} \geq x y$, that is $(x-y)^{2} \geq 0$. We have equality in this inequality if and only if $x = y$.
Applying the inequality above we get
$a^{3} + b^{3} \geq a b(a+b)$, $b^{3} + c^{3} \geq b c(b+c)$, $c^{3} + a^{3} \geq c a(c+a)$.
Adding these inequalities we obtain (1). We have equality in our inequality if and only if $a = b = c$.