I remember $\mathcal{O}_{X}$ is a sheaf of the open sets of $X$, in this it looks like it is a sheaf of modules; but in the superscript of the sequence what are the direct summations over indices $I,J$? What is the superscript $n$ used in the second property of a coherent sheaf? I just don't have experience with superscripts in denoting a sheaf yet so would someone fill me in please?
In this case, $\mathcal{O}_X^{\oplus I}$ is just the direct sum of copies of $\mathcal{O}_X$ indexed by the set $I$. So, for instance if $I=\{1,\ldots, k\}$, then $$\mathcal{O}_X^I=\underbrace{\mathcal{O}_X\oplus\cdots \oplus \mathcal{O}_X}_{k\:\text{times}}.$$ However, we can take $I$ to be an infinite indexing set in which case this becomes a direct sum with terms bijective with $I$. $\mathcal{O}_X^n$ is another way of writing $$ \underbrace{\mathcal{O}_X\oplus\cdots \oplus \mathcal{O}_X}_{n\:\text{times}} $$ when $n\in \Bbb{N}$. Indeed, you should note that for quasicoherent sheaves, we do not require that the indexing sets $I$ and $J$ in your image be finite. For coherent sheaves, we do stipulate that $I$ and $J$ are finite.
This notation is not special to sheaf theory. You could do the same thing with rings, modules, etc.