Why is self-information defined the way it is?
Though it is more common to discuss axiomatic derivation of the expectaion of $I(\cdot)$, that it, the entropy function (to which, several sets of axioms exist, see here), for the self information $I(\cdot)$, we usually care about the following properties (as defined by Shannon):


If the event happens with certainty, there is no information in it. Thus, if $p=1$, $I(p)=0$
Furthermore, $I(p)$ must be decreasing with $p$: the more likely the event is, the less information it gives us (or alternatively, the more surprising the event is, the more information its occurrence contains)
As you mentioned, we want that the information of two independent events will add up: $I(p_xp_y)=I(p_x)+I(p_y)$. The event of both $X$ and $Y$ occurring has probability $p_x\cdot p_y$ if independent.


These axioms lead to $I(p) =\log \frac1p$, up to a normalization (e.g., choosing the basis of the logarithm).