Intuition surrounding units in $R[x]$
There is a natural way to inject $R$ into $R[x]$ by sending an element $a$ to the polynomial $a$ (so the coefficient of $x^i$ is 0 for all $i>0$. Now it should be easy to see that the image of a unit under this injection is again a unit. On the other hand, if some polynomial $p$ is a unit in $R[x]$ then there is another polynomial $q$ such that $pq = 1$, but if $p$ has degree $>0$ then so does $pq$ since this is over a domain. Thus, if $p$ is a unit then $p$ is actually constant, and clearly the only constants that are units are the ones that are already units in $R$.