Here is an argument using way more abstract machinery than Jonas Meyer's.
First, note by the Hahn-Jordan decomposition theorem that any finite signed measure $\mu$ on $[0,1]$ can be written $\mu = \mu^+ - \mu^-$ for positive measures $\mu^+$,$\mu^-$. Thus in fact we have $\int f_n \,d\mu \to 0$ for every finite signed measure $\mu$.
The Riesz representation theorem states that the dual of the Banach space $C([0,1])$ is precisely the space of finite signed measures. So in fact our condition is that $f_n \to 0$ weakly in $C([0,1])$.
Now it follows from the uniform boundedness principle that a weakly convergent sequence in a normed space is bounded in norm. Thus we have $\sup_n \|f_n\|_\infty < \infty$ which is to say that the $f_n$ are equibounded.
(Sketch of the last step: for each $n$, the map $\ell_n(\mu) = \int f_n\,d\mu$ is a continuous linear functional on $C([0,1])^*$. By assumption, we have for each $\mu$ that $\sup_n |\ell_n(\mu)| = \sup_n \left|\int f_n\,d\mu\right| < \infty$ since $\{\int f_n\,d\mu\}$ is a convergent sequence of real numbers and hence bounded. So by the uniform boundedness principle, $\sup_n \|\ell_n\|_{C([0,1])^{**}} < \infty$. But by the Hahn-Banach theorem, we have $\|\ell_n\|_{C([0,1])^{**}} = \|f_n\|_\infty$, i.e. the natural map $C([0,1]) \to C([0,1])^{**}$ is an isometry.)