Already posted here and here.

Perhaps the simplest way will be to assume WLOG $c\ge a,b$. Then $\frac {a}{b+c}+\frac {b}{c+a}+\frac {c}{a+b}\le \frac {a}{b+a}+\frac {b}{b+a}+\frac {c}{a+b}<2$.

As theses are sides of a triangle , let $a = x+y$ , $b = y+z$ , $c = z+x$ and using homogeneity set $x+y+z = 1$. The inequality is then to show : $\frac{1-x}{1+x} + \frac{1-y}{1+y} + \frac{1-z}{1+z} < 2$. Note that $x \in (0,1) \implies 1 + x > 1 \implies \frac{1-x}{1+x} < 1 - x$. Sum that across $x,y,z$ to get the desired inequality.

Let me know if you have got some questions