$\Longrightarrow:$ Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a continuous function. Assume that $f \in L^1 (\mathbf{R})$. Then $\int_\mathbf{R} |f| dm < \infty$ by definition. Also we know that every continuous function on the interval $[a,b]$ is Riemann integrable. Hence since $|f|$ is continuous on $[a,b]$, $|f|$ is Riemann integrable on the interval $[a,b]$. We also know that
$$\int_{[a,b]} |f| dm = \int_a^b |f| dx.$$
Also since $|f|$ is non-negative, we obtain the following the inequality
$$\int_{[a,b]} |f| dm \leq \int_\mathbf{R} |f| dm \text{ for any } a,b \in \mathbf{R}.$$
Note that the value of the integral is increasing. Therefore by Monotone Convergence Theorem, we have:
$$\lim_{a \rightarrow - \infty, b \rightarrow \infty} \int_a^b |f(x)| dx=\lim_{a \rightarrow - \infty, b \rightarrow \infty} \int_{[a,b]} |f(x)| dm = \int_\mathbf{R} |f| dm.$$
$\Longleftarrow:$ On the other hand, suppose that the limit $\lim_{a \rightarrow - \infty, b \rightarrow \infty} \int_a^b |f(x)| dx$ exists and it is finite. We know that
$$\int_{[a,b]} |f| dm = \int_a^b |f| dx.$$
Taking the limit of both sides, we obtain
$$\lim_{a \rightarrow - \infty, b \rightarrow \infty} \int_a^b |f(x)| dx=\lim_{a \rightarrow - \infty, b \rightarrow \infty} \int_{[a,b]} |f(x)| dm= \int_\mathbf{R} |f| dm,$$
as desired.
In order to prove, $\int_\mathbf{R} f(x) dm = \lim_{a \rightarrow - \infty, b \rightarrow \infty} \int_a^b f(x) dx$, we write $f$ as a difference of two non-negative functions $f=f^+ - f^-$. Since $f^+$ and $f^-$ are both non-negative, the result we proved holds. Hence $$\int_\mathbf{R} f^+(x) dm = \lim_{a \rightarrow - \infty, b \rightarrow \infty} \int_a^b f^+(x) dx$$ and $$\int_\mathbf{R} f^-(x) dm = \lim_{a \rightarrow - \infty, b \rightarrow \infty} \int_a^b f^-(x) dx$$ and therefore we conclude that $$\int_\mathbf{R} f(x) dm = \lim_{a \rightarrow - \infty, b \rightarrow \infty} \int_a^b f^+(x) - f^-(x) dx,$$ as desired.
