"The Mexican War may accurately be blamed for causing the Civil War." Why?

It is (at least arguably) accurate to say that the Mexican War caused the Civil War because the Mexican War upset the status quo on slavery.  By doing so, it led to more conflict between the North and the South.  This conflict eventually led to the Civil War.

Before the war with Mexico, the North and South had come to an uneasy compromise on the issue of slavery.  This had been put in place in 1820 with the creation of the Missouri Compromise.  People knew which areas could have slaves and which could not.  Many people were unhappy with the situation, but not enough to cause serious trouble.  The war with Mexico changed this.

First, the war itself caused discord between the North and the South.  Many people in the North felt that the war was being fought in order to take more territory for slavery.  For this reason, many Northern members of Congress supported the Wilmot Proviso.  This proposal stated that slavery should be banned from any land taken from Mexico in the war.  Of course, the South opposed the Proviso vigorously.  Thus, the war itself brought about some animosity between the two regions.

Once the war was over, things only got worse.  The US took huge areas of land from Mexico.  Those areas had to be organized into territories.  This brought up the issue of whether the territories made out of the Mexican Cession would be free or slave.  In other words, the war forced the North and South to come into conflict over what areas would and would not have slavery.

This conflict led to the Compromise of 1850, which was meant to calm things down just as the Missouri Compromise had three decades before.  The Compromise of 1850 was largely successful, but one part of it was terribly controversial.  This was the Fugitive Slave Act, which made it much easier for escaped slaves to be returned to their owners.  Perhaps more importantly, it required private citizens to help capture escaped slaves and it set up a process that would make it easy for any black person, even a free one, to be sent South.  The North resented this law and the South was unhappy that the North was so resistant to returning slaves.  This helped pull the two regions apart.

The Fugitive Slave Act also helped influence Harriet Beecher Stowe to write Uncle Tom’s Cabin.  This book was influential enough that Abraham Lincoln is supposed (though this may be apocryphal) to have said to Beecher Stowe that it caused the Civil War.  The book caused conflict between the North and South because it increased anti-slavery feeling in the North.

We can even say that the Mexican War helped bring about the Kansas-Nebraska Act.  This law was created in order to build a transcontinental railroad.  If the US had not taken the Mexican Cession, there would have been no need for such a railroad.  The Kansas-Nebraska Act opened Kansas up to slavery if the people wanted it, thus abrogating the Missouri Compromise.  The discord over this law, and the violence of “Bleeding Kansas” helped deepen the enmity between North and South.  For this reason, the Kansas-Nebraska Act is generally seen as a major cause of the Civil War (it even gave John Brown his start in anti-slavery violence).

In short, the Mexican War reopened the issue of slavery in the United States.  It forced the North and South to try to agree again on what areas should and should not have slavery.  In doing this, it helped bring about many of the events that led directly to the Civil War.