Answer
We conclude that $f(x)=0$ for every $x\in[a,b].$
Work Step by Step
If $f$ is continuous on $[a,b]$ then also $|f|$ is continuous on $[a,b]$ and also nonnegative on $[a,b]$. If the integral of a nonnegative continuous function on an interval of nonzero length is equal to zero, the only possibility is that that function is equal to zero everywhere on that interval. If we suppose otherwise then there must be regions bounded by the graph of the function and the $x$ axis both above and below the $x$ axis such that they cancel out but this is impossible because the function is nonnegative and no pieces of the mentioned region can be below the $x$ axis. So we conclude that $|f(x)|=0$ for $x\in[a,b]$ and that means that also $f(x)=0$ for $x\in[a,b].$
