First, when you say "dense" you need to specify with respect to what norm. I'm assuming you mean the uniform norm (the sup norm).
That said, the statement is false. To see that, take $f(x)=c$, where $c\neq0$. Clearly $f\in C(\mathbb{R})\cap L^\infty(\mathbb{R})$ (continuous and bounded). Now, for every function $\varphi\in C^\infty_0$, we have $||\varphi-f||\geq c$. Hence, there is no function in $C^\infty_0(\mathbb{R})$ that is arbitrarily close to $f$, which means that $C^\infty_0(\mathbb{R})$ is NOT dense in $C(\mathbb{R})\cap L^\infty(\mathbb{R})$.