I think it's well known that if $X\subset\mathbb{P}^3$ is a smooth cubic surface and we take the projection $\pi: X\rightarrow \mathbb{P}^2$ from a point off the surface, then it's branched over a sextic curve with 6 cusps.

Why is this true? In particular, I'm not seeing the 6 cusps.

Example of a statement of the fact: *the Problem of Existence of Algebraic Functions* by Zariski (1929), pg 320

lie on the same conic(for instance, this is shown by the argument in abx's answer). $\endgroup$