Imagine the plane $(x,a)$ where the equation $f(x,a)=0$ gives a curve $C$. We project the curve onto the $a$-axis. Branch points are the points on the $a$ axis over which the projection "is not so nice". You can imagine e.g. the points over which the projection "folds", e.g. if $f(x,a)=x^2-a$ then $a=0$ is such a point.
Now you have to imagine, though, that $a$ and $x$ are complex numbers (and hence $a$ is a point in a plane, and $C$ is 2-dimensional). Over a general point $a$ the equation $f(x,a)=0$ has $n$ different solutions (for $x$), where $n$ is the degree of $f$ as a polynomial in $x$ (in your case $n=3$). We have $n$ different solutions iff $f(x,a)=0$ and $df(x,a)/dx=0$ (for this given $a$) don't have a common root. Over such points $C$ covers the plane of $a$'s in $n$ sheets. When, however, $f(x,a)=0$ and $df(x,a)/dx=0$ do have a common root, the number of roots of $f(x,a)=0$ is less than $n$, so over these points $a$ some of the sheets meet. These are the branch points. In fact, what happens, if you go around such a point then you would see that you get from one sheet (also called one "branch") to another. That's why these points are called branching.
(I skipped some unimportant technical details)