If $V$, $E$ and $F$ is respectively the number of vertices, edges and faces in a maximally triangulated graph, then why do we have $3F = 2E$?
Every face is a triangle (hence, the name, "maximally triangulated graph"). Every face is thus composed of three edges (or sides). So, if we have $F$ faces, then we may be tempted to say that there are $3F$ edges. This is not the case because an edge contributes to exactly $2$ faces. So, in $3F$ we are counting every edge twice. So, $\frac{3F}{2} = E \iff 3F = 2E$.