Consider the normal distribution. We know that $$p(x| \mu, \sigma^2) = \frac{1}{\sqrt{2 \pi \sigma^{2}}} e^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}} $$
The kernel is $$ p(x| \mu, \sigma^{2}) \propto e^{-\frac{(x-\mu)^{2}}{2 \sigma^{2}}} $$ omitting the part that isn't a function of $x$. Why write $p(x|\mu, \sigma^{2})$ like this?