Keep in mind that Matt didn't fully explain WAR there. Every one of the five lines in his first paragraph is, on the face of it, a black box.
Now, there are formulas behind each of those black boxes - formulas about which analysts often disagree (which is why BR's WAR differs from FG's WAR, and presumably other folks' WAR as well). How are batting runs calculated (for each "flavor" of WAR)? How are baserunning, double play, fielding runs calculated? How is relative positional value calculated?
To really understand, you need to drill back to the actual counting statistics that drive all these calculations. And then you need to decide which of the more subjective raw statistics (i.e., things based on judgment calls by scorers, as opposed to objective events) you find more believable (again, this is one of the places where different versions of WAR disagree).
When all that is done, you need to remember that WAR - all its versions - is based on counting stats, not rate stats. WAR can tell you how many wins above replacement a player provided in a given season (subject to the uncertainties of its underlying counting stats), but it doesn't really tell you how efficient the player was, because it's not measured per plate appearance, or per inning in the field or on the mound. It can give you an idea what to expect in the (near) future from a player, but only if that player actually gets playing time equivalent to the season it's been calculated from.
Personally, I think that WAR tries to do too much - tries to reduce too many variables to a single number. In particular, I don't trust the defensive stats; there's relatively little agreement about which defensive measures are better, and (IMHO) enough year-to-year variability to suggest to me that something is just...broken.
I see WAR as kind of like the Standard Model of cosmology. It's useful, it can provide insight... but at the same time, it's clearly not "right." There are issues, things that are wrong, that we don't quite understand. So my approach is, "use it, but don't quite believe it."