From Mac Lane's Category Theory, what exactly is being assumed in the statement of this theorem? What does the enclosed red box in the below picture mean?
I can see that $C$ has equalizers of all pairs of arrows means every pair of arrows in $C$ has an equalizer, but I don't understand what the next part is saying. Can someone EXPLITICTLY explain what this means?
We take products of objects in a category indexed by a set. This is all he is saying: if products always exist for index sets of certain sizes, you will have all colimits in the shape of $J$ (assuming coequalizer stuff). And, of course, you might have all products, not just those of a certain size, but that is just gravy and does not violate any sufficient condition he gives.
Given an index set $I$, we may consider a set of objects of $\mathcal{C}$ indexed by $I$. This would be a collection of objects $\{X_i : i \in I\}$, each $X_i$ an object of $\mathcal{C}$. We may then speak of (or, ask for the existence of) the product $\prod_{i \in I} X_i$. This is a "product indexed by $I$."
Mac Lane is assuming that ALL such products exist whenever the index set is $\mathrm{obj}(J)$ or $\mathrm{arr}(J)$. This is really an assumption on the (roughly, modulo cardinalities) number of factors in such products. If your object set is $\{a, b, c\}$ it certainly does not matter if you index a product of three terms as $X_a \times X_b \times X_c$ or $X_1 \times X_2 \times X_3$: it's still a product of three objects.