If $S_n=X_1+\cdots+X_n$ with $(X_n)$ independent and $X_n$ with density $f_n$, then $ \mathrm P(S_n\leqslant t\mid X_n=x)=\mathrm P(S_{n-1}\leqslant t-x\mid X_n=x)=\mathrm P(S_{n-1}\leqslant t-x)=F_{n-1}(t-x), $ where the second equality stems from the fact that $S_{n-1}$ and $X_n$ are independent. Integrating this over $x$, one gets $ F_n(t)=\mathrm P(S_n\leqslant t)=\int\mathrm P(S_n\leqslant t\mid X_n=x)\mathrm dP_{X_n}(x)=\int F_{n-1}(t-x)f_n(x)\mathrm dx. $ In the case at hand $S_0=0$ and $S_1=X_1$ hence one can start the recursion at $F_0(t)=\mathbf 1(t>0)$ or at $F_1(t)=\mathrm P(X_1\leqslant t)=\int\limits_0^tf(x)\mathrm dx$. And all the $X_k$ are almost surely positive hence $f_n(x)=0$ for $x\leqslant0$ and $F_{n-1}(t-x)=0$ for every $x\geqslant t$, thus the integral giving $F_n(t)$ may be restricted to $0\leqslant x\leqslant t$.
More generally, if $W=U+V$ for independent random variables $U$ and $V$ and if $U$ has CDF $G$ defined by $G(t)=\mathrm P(U\leqslant t)$ and $V$ has density $h$, then the CDF of $W$ is $ \mathrm P(W\leqslant t)=\int G(t-x)h(x)\mathrm dx. $