Example 8 If $a+b+c=1$, then the maximum value of $\sqrt{3 a+1}+$ $\sqrt{3 b+1}+\sqrt{3 c+1}$ is $\qquad$ . (Example 10 from [1])
By Cauchy-Schwarz inequality, we have $(\sqrt{3 a+1}+\sqrt{3 b+1}+$ $\sqrt{3 c+1})^{2} \leqslant\left(1^{2}+1^{2}+1^{2}\right)[(3 a+1)+(3 b+1)+$ $(3 c+1)]$,

Combining $a+b+c=1$, we get $\sqrt{3 a+1}+\sqrt{3 b+1}+$ $\sqrt{3 c+1} \leqslant 3 \sqrt{2}$, so the maximum value we seek is $3 \sqrt{2}$. (At this point, $a=$ $\left.b=c=\frac{1}{3}\right)$