How is the idea of Manifest Destiny reflected in the annexation of Texas?
Texas was a necessary first piece of the land needed to fulfill the "destiny" of spreading from "sea to shining sea."
Manifest destiny was a belief and a concept that the US was meant to spread to the Pacific Ocean. Given the land claims of the US, Texas was vital in achieving this.
Beyond that, Stationing troops in Texas in land that was disputed with Mexico was seen as an act of war by Mexico. The skirmish that resulted allowed President Polk to go to Congress requesting a declaration of war on Mexico, which he got.
The Mexican American War, of course, was also necessary to complete this destiny. The US had approached Mexico to purchase the land but Mexico was not interested in the deal. So Polk's maneuver to station the troops is seen by some as goading Mexico into war in order to win the land that some thought they were destined to have.