I pick up your remarks about sheaves. Indeed, the sheaf condition is a very good example to get a geometric idea of a limit.
Assume that $X$ is a set and $X_i$ are subsets of $X$ whose union is $X$. Then it is clear how to characterize functions on $X$: These are simply functions on the $X_i$ which agree on the overlaps $X_i \cap X_j$. This can be formulated in a fancy way: Let $J$ be the category whose objects are the indices $i$ and pairs of such indices $(i,j)$. It should be a preorder and we have the morphisms $(i,j) \to i, (i,j) \to j$. Consider the diagram $J \to Set$, which is given by $i \mapsto X_i, (i,j) \mapsto X_i \cap X_j$. What we have remarked above says exactly that $X$ is the colimit of this diagram! In a similar fashion, open coverings can be understood as colimits in the category of topological spaces, ringed spaces or schemes. It's all about gluing morphisms.
Now what about limits? I think it is important first to understand limits in the category of sets. If $F : J \to Set$ is a small diagram, then we can consider simply the set of "compatible elements in the image" of $F$, namely
$X = \{x \in \prod_j F(j) : \forall i \to j : x_j = F(i \to j)(x_i)\}$.
A short definition would be $X = Cone(*,F)$. Observe that we have projections $X \to F(j), x \mapsto x_j$ and with these $X$ is the limit of $F$. Now the Yoneda-Lemma or just the definition of a limit tells you how you can think of a limit in an arbitrary category: That $X$ is a limit of a diagram $F : J \to C$ amounts to say that elements of $X$ .. erm we don't have any elements, so let's say morphisms $Y \to X$, naturally correspond to compatible elem... erm morphisms $Y \to F(i)$. In other words, for every $Y$, $X(Y)$ is the set-theoretic limit of the diagramm $F(Y)$. I hope that this makes clear that the concept of limits in arbitrary categories is already visible in the category of sets.
Now let $X$ be a topological space and $O(X)$ the category of open subsets of $X$; it's an preorder with respect to the inclusion. Thus a presheaf is just a functor $F$ from $O(X)^{op}$ to the category of sets (or which suitable category you like). Now open coverings can be described as certain limits in $O(X)^{op}$, i.e. colimits in $O(X)$, as above. Observe that $F$ is a sheaf if and only if $F$ preserves these limits: If $U$ is covered by $U_i$, then $F(U)$ should be the limit of the $F(U_i), F(U_i \cap U_j)$ with transition maps $F(U_i) \to F(U_i \cap U_j), F(U_j) \to F(U_i \cap U_j)$, i.e. $F(U)$ consists of compatible elements of the $F(U_i)$, meaning that the elements of $F(U_i)$ and $F(U_j)$ restrict to the same element in $F(U_i \cap U_j)$. Thus we have a perfect geometric example of a limit: the set of sections on an open set is the limit of the set of sections on the open subsets of a covering.
Somehow this view takes over to the general case: Let $F : J \to Set$ be a functor. Regard it as a presheaf on $J^{op}$, and the map induced by $i \to j$ in $J^{op}$ as a restriction $F(j) \to F(i)$. Also call the elements of $F(i)$ sections on $i$. Then the limit of $F$ consists of compatible sections. Since I've been learning algebraic geometry, I almost always think of limits in this way.
Finally it is important to remember that limit is just the dual concept of colimit. And often algebra and geometry appear dually at once, for example sections and open subsets in sheaves. If $(X_i,\mathcal{O}_{X_i})$ are ringed spaces and you want to find the colimit, well you can guess that you have to do: Take the colimit of the $X_i$ and the limit of the $\mathcal{O}_{X_i}$ (pullbacked to the colimit).
"...the sheaf condition on a presheaf can be expressed as stating that the contravariant functor takes colimits to limits"
This is not correct. The reason is that the index category can be rather wild and colimits in preorders don't care about that. In detail: Let $U : J \to O(X)^{op}$ be a small diagram. Then the limit is just the union $V$ of $U_j$. Thus $F$ preserves this limit iff sections on $V$ are sections on the $U_j$ which are compatible with respect to the restriction morphisms given by $U$. If $J$ is discrete and $U$ maps everything to the same open subset $V$ of $X$, then the compatible sections are $F(V)^J$, which is bigger than $F(V)$.
"... I have a copy of MacLane's "Categories for the Working Mathematician," but whenever I pick it up, I can never seem to get through more than two or three pages (except in the introduction on foundations"
I think this book is still one of the best introductions into category theory. It can be hard to grasp all these abstract concepts and examples, but it gets easier as soon as you get input from other areas where category theoretic ideas are omnipresent. Your example about gluing morphisms illustrates this very well.
If we interpret Voevodsky's first claim broadly, there have been several high-profile results that the mathematical community has had great difficulty verifying, e.g., Perelman's proof of the geometrization conjecture, the classification of finite simple groups, Hales and Ferguson's proof of the Kepler conjecture, and Mochizuki's proof of the abc conjecture.
Perelman's proof is now accepted but it took years for the community to validate it.
The classification of finite simple groups is now regarded as not having been completely proved until Aschbacher and Smith's work in 2004, but for many years the generally accepted date for the completion of the proof was 1983 and it took a while for the quasithin case to be generally acknowledged as a serious gap.
Hales is working on the Flyspeck project, which is a tacit acknowledgment that the original proof was too hard for the community to independently verify and that formal mechanized proofs are the way to go.
Mochizuki's proof is still in the process of being verified almost two years after he made the proof public, with no closure yet on the horizon.
This is already an impressive list in my opinion and does not even touch on less famous results, or results that generated controversy such as Hsiang's proof of the Kepler conjecture or the original proof of the four-color theorem. (Ironically, of course, many people initially were uncomfortable with the proof of the four-color theorem because computers were involved, whereas today Voevodsky is uncomfortable unless computers are involved—albeit in a different way.)
Best Answer
The answer is essentially given in the comments, so let me summarize:
It is a frequent situation that one has to cite an exercise.
It is legitimate. (Polya-Szego is cited > 1400 times according to Mathscinet)
The best thing is to cite a place where the statement is proved, but if you cannot find such a place, citing an exercise is the second best choice.
You can solve the exercise in your paper, or not solve (depending on the difficulty of the exercise and space limitations and other considerations).
And finally my own recommendation: When you refer to an exercise, solve it yourself, no matter whether you include a solution to your paper or not.
Similar considerations apply to handbooks, like Tables of Integrals, etc. They are essentially made for this purpose, but there are sometimes mistakes, not frequently. (Gradshtein-Ryzhik is cited > 2200 times according to Mathscinet, Abramowitz-Stegun 1740.)