For fixed $x\in\mathbb{R}$, let $y=nt$, then $$\tag{1}n \displaystyle\int_0^1 f(x+t) g(nt) dt=\int_0^nf(x+\frac{y}{n})g(y)dy=\int_0^\infty \chi_{[0,n)}(y)f(x+\frac{y}{n})g(y)dy.$$
Here $\chi_{[0,n)}$ is the characteristic function:
$$\chi_{[0,n)}(y)= \begin{cases}
1, & y\in[0,n)\hbox{;} \\
0, & \hbox{otherwise.}
\end{cases}
$$
For $y\in[0,n)$, $x\leq x+\displaystyle\frac{y}{n}< x+1$. Since $f$ is continuous on $[x,x+1]$, let $M=\max_{z\in[x,x+1]}|f(z)|<\infty$, then
$$\big|\chi_{[0,n)}f(x+\frac{y}{n})g(y)\big|\leq M|g(y)|\in L^1([0,\infty)).$$
Therefore, by Dominated convergence theorem, we have by $(1)$
$$\lim_{n\rightarrow\infty}n \displaystyle\int_0^1 f(x+t) g(nt) dt=\int_0^\infty \lim_{n\rightarrow\infty}\chi_{[0,n)}(y)f(x+\frac{y}{n})g(y)dy$$
$$=
\int_0^\infty f(x)g(y)dy=f(x)\int_0^\infty g(y)dy=f(x)$$
Here we have used the assumption that $f$ is continuous, which implies that $\lim_{n\rightarrow\infty}f(x+\frac{y}{n})=f(x)$ for all $y\in[0,n)$.