Suppose that $d$ is a pseudometric generating the topology of $X$, and for $x,y\in X$ let $x\sim y$ iff $d(x,y)=0$. Then $X/\\!\sim$ is metrizable by $\bar{d}\big([x],[y]\big)=d(x,y)$, where $[x]$ is the $\sim$-class of $x$. In effect we’ve simply collapsed the topologically indistinguishable points of $X$. ($X/\\!\sim$ is the Kolmogorov quotient of $X$.)
Turn that around, and you see that pseudometrizable spaces are just metrizable spaces in which some of the points may have been ‘fattened up’ to sets of pairwise topologically indistinguishable points. Take just about any characterization of metrizability and weaken it by not requiring the space to be $T_0$, and you’ll have a characterization of pseudometrizability.