A large part of the Lost Cause's mythology is that the CSA was only defeated by the Union because of the Union's overwhelming advantages in population, technological advancement, and productive capacity, while the Confederacy had much better leaders and braver soldiers. Given how much of the Lost Cause is inaccurate, I have trouble believing this point. However, I have not read any evidence to the contrary so I would like to hear from the experts on this topic.
Largely by attrition because your basic premise is correct. The Union was more powerful than the confederacy in almost every concievable way. It had the better infrastructure by far (with about 2,5 times the railroad mileage compared to the CSA), the greater recruitable population (4,5 times that of the confederacy if we define "recruitable" as caucasian male aged 18 to 45), the greater GNP (3 times that of the confederacy), the list goes on.
Now, military history has a nasty tendency to get pulled into a discussion of tactical and strategic decision of officers since that is easy to study. However those decisions in turn are the result of politics, logistics and information. On a very fundamental level, war is just a pissing contest of economy. The one who can piss away the most usually wins.
The latter I find to be particularly true for the US civil war due to the lack of professional soldiers, competent officers and military tradition on both sides. The US civil war was very disorganized compared to other contemporary conflicts. I will try to elaborate a little on why I make this claim.
The Civil War was fought using tactics considered (by Europeans) out dated by the mid 1800's. Napoleonic era maneuver units with large blocks of men in tight formations. This was partly the result of lacking modern military knowledge but also the result of officers and men being amateurs. Large formations require less training and discipline than small unit tactics and rifle chains.
To make matters worse, both sides tended to use these large units in the worst possible way. The strength of a maneuver unit is that it can bring great weight to a charge. Napoleon proved this time and time again. If you charge a formation with an attack column the odds are greatly in your favour. However, if you do what was done in almost every battle of the US Civil War, that is, march your tightly packed units under artillery fire against an entrenched enemy and fucking stop, 50-100 yards out, right in the middle of the defenders kill zone, and try to out-shoot them, you're going to have a really really bad time. Even when this actually worked, or when an actual charge was carried out, the attackers were usually too depleted and exhaused to follow through which means most battles were, by European standards, inconclusive. The loser could fall back, reinforce and be back on the field as quickly as the "winner" could.
Another factor that made for even more inconclusive battles is the lack of competent cavalry on either side. Sure, there were plenty of skilled horsemen, especially in the south, but precious little knowledge, interest or discipline to make a competent Napoleonic era cavalry force (which is what they would have needed since they were fighting a napoleonic era infantry war). Cavalry was used more like dragoons than actual cavalry. The preformed well enough as raiders, scouts and mobile infantry but generally did not preform at all in either of their primary roles, charging infantry and, most importantly, shattering and pursuing a retreating enemy.
In lieu of decisive victories that actually shattered an enemy army we're left with what we had. A slow and bloody series of battles the ended only when armeis were out-maneuvered and cornered.
I left out the part about the south having "better leaders and braver soldiers" because:
A) I see no decent, objective way to quantify that.
B) My personal opinion is that Grant was by far the best leader of the war.
Regarding the point about "braver soldiers," Shelby Foote once shot that down in a pretty interesting way. While Confederate troops were supposed to have more elan, he felt that all you needed to do was to look at the repeated attempts to take Marye's Heights by the Union during the Battle of Fredricksburg, of which there were 14, and all of which failed. Soldiers who participated could watch their comrades be slaughtered, and somehow those northern soldiers found the courage to keep trying. That isn't particularly unique, either: you can find plenty of examples (on both sides) of soldiers going into circumstances where there was an excellent chance they'd be killed.
Or, from another perspective, look at desertion rates. The last reckoning I saw estimated that around one in ten Union soldiers deserted, and one in nine Confederate soldiers deserted. Among Virginian soldiers, the desertion rate was between 10% and 15%; the average in the Union varied from 9% to 12%. That isn't to say that Virginians were more cowardly, because the desertions spiked once it was obvious that the war was lost (I don't think it's reasonable to fault a person for refusing to die for a lost cause or a bad idea, but whatevs, people accuse me of being selfish), and the Union played it smart and actually encouraged Confederates to desert with lenient amnesty policies, but that idea that they were fanatically brave and willing to fight to the bitter end doesn't really play.