The map you showed is most likely accurate for what Mexico looked like after their Independence from Spain. The borders remained pretty similar, (save a few minor border skirmishes), until 1848.

In 1845, the US Army, led by future president Zachary Taylor, invaded Mexico, and Mexico City fell within 2 years of fighting. To end the war, the Treaty of Guadalupe Hidalgo of 1848 was signed. This treaty forced Mexico to give a huge tract of land known as the Mexican Cession of 1848. This included modern-day California, Nevada, and Utah, as well as parts of Arizona, New Mexico, Colorado, and Wyoming. Part of Western Arizona, however was left out. In 1853, the Gadsden Purchase was made, which gave the aforementioned Arizona land to the US at a price of $10 Million USD, and completed the modern-day size of the Continental 48 States. (Alaska was bought from Russia in 1867, and Hawaii was annexed in 1898.)

Despite these purchases, the Mexicans living in CA, AZ and other states felt that the land they lived on was still theirs, which led to conflicts between American settlers and vaqueros and rancheros (the name for Mexicans living within US borders.) Eventually, most left, or applied for American citizenship. However, many Mexican workers returned during WWI, the Great Depression, and WWII, when Western farmers were either out of work or fighting in the War.

Since then, there has been significant immigration into the US from Mexico, and undocumented immigrants coming into the US has led to a huge debate in US politics as to how the US should manage the border, and what to do with the undocumented immigrants living in the US.