Solution:
By the Cauchy-Schwarz inequality it is
$$\begin{aligned} & \left(\frac{a^{2}}{a^{2} + ab} + \frac{b^{2}}{b^{2} + bc} + \frac{c^{2}}{c^{2} + ca}\right)\left((a^{2} + ab) + (b^{2} + bc) + (c^{2} + ca)\right) \geq (a + b + c)^{2} \\ \Rightarrow & \frac{a^{2}}{a^{2} + ab} + \frac{b^{2}}{b^{2} + bc} + \frac{c^{2}}{c^{2} + ca} \geq \frac{(a + b + c)^{2}}{a^{2} + b^{2} + c^{2} + ab + bc + ca} \end{aligned}$$
So it is enough to prove $\frac{(a + b + c)^{2}}{a^{2} + b^{2} + c^{2} + ab + bc + ca} \geq \frac{a + b + c}{2}$, that is to prove
$$2(a + b + c) \geq a^{2} + b^{2} + c^{2} + ab + bc + ca$$
Substituting $a^{2} + b^{2} + c^{2}$ for $a + b + c$ into the left hand side we wish equivalently to prove
$$a^{2} + b^{2} + c^{2} \geq ab + bc + ca$$
But $a^{2} + b^{2} \geq 2ab$, $b^{2} + c^{2} \geq 2bc$, $c^{2} + a^{2} \geq 2ca$ which by addition imply the desired inequality.