How can the direction of the electric field of a negative point charge be going in towards itself (radially), when the charge itself is generating the electric field? Is it generated at infinity then? Because if it's generated by the charge then it cannot go further in towards itself?
In case of positive charge, it is understandable that the field is being generated at the charge (at it's surface to be precise) and is going outwards radially.
Best Answer
There is no "going" going on in field-line diagrams. The direction of the field lines indicates, by convention, the direction of the electrostatic force experienced by a positive test charge at that location.
Field lines do not indicate the 'flow' of any physical quantity, and there is nothing being 'generated'; instead, all you have is a force field, and ways to study and analyze it. This extends to the concept of electric flux (i.e. for a given surface $S$, the integral $\iint_S\mathbf E\cdot\mathrm d\mathbf S$): we call it 'flux' by analogy, but there's nothing at all actually 'flowing'; instead, it is just one more tool to understand and analyze the force field and the laws that govern it.
For more on field lines, see Why does the density of electric field lines make sense, if there is a field line through every point?.