What is the Gelfand-Naimark representation of functions that don't vanish at infinity?

Since $C_b(Y)$ has a unit, if it is isometrically $^*$-isomorphic to $C_0(X)$ then $C_0(X)$ must also have a unit, so $X$ is compact. This is intuitively why $X$ will need to be some sort of compactification of $Y$. To be exact, we will get the Stone Cech compactification.

The Stone Cech compactification $\beta Y$ of $Y$ comes with an embedding $\Delta:Y\to\beta Y$ such that $\Delta(Y)$ is dense in $\beta Y$. The important property is that $f\mapsto f\circ\Delta:C(\beta Y)\to C_b(Y)$ is an isometric $^*$-isomorphism. Since $\beta Y$ is by definition also compact, we have $C_0(\beta Y)=C(\beta Y)$, so $C_b(Y)$ is isometrically $^*$-isomorphic to $C_0(\beta Y)$.

Do note that $Y$ is required to be completely regular for the Stone Cech compactification to be well-defined. I do not know about the case where $Y$ is not completely regular.