I wrote up a whole explanation of how to calculate WAR and then my computer crashed and ate it. Maybe I'll try again tomorrow.
Anyway, regarding Trout in particular: he does so well by this metric because he's good at everything. He has a good batting average, a good walk rate, he hits for power, he steals bases, he plays an important defensive position, he even does little things like advancing from first to third on a single really well. Being a well-rounded player is a big advantage. (Willie Mays, the classic well-rounded player, is 5th all-time in WAR, behind Ruth, Cy Young, W. Johnson, and Bonds.)
On the Trout/Pujols comparison: there are three big reasons for the difference. One is what I mentioned above, Trout is more well-rounded than Pujols ever was. Pujols may have been the better hitter, but the other stuff makes up for the difference in their bats.
The second is that Trout is a center fielder and Pujols was (mostly) a first baseman. WAR gives players a positional adjustment, weighted to the number of games that they play at each position. First basemen, left fielders (and especially DHs) get a negative adjustment; guys who plays hard defensive positions get a bonus. Trout has been getting about +0.3 WAR bonus each year for playing center, Pujols lost about 0.8 per year for playing first. The rationale for this is that its easier to find a guy who hits well and plays first base than one who hits well and plays a tough defensive position. Clearly a shortstop who could hit like Pujols would be more valuable than a first baseman that could hit like Pujols, and the positional adjustment is what lets us account for that fact.
Finally, Pujols' prime seasons were during a more offense-heavy era. There's about a 20 point difference in league average OPS between 2001 and 2018. Because it was easier to put up impressive offensive numbers in 2001, that performance isn't as valuable as the same performance would be in 2018.
That's all I've got time for right now. Hopefully I manage to recreate the post on calculating WAR tomorrow.
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