People moved to Texas for easy access to land of their own. The concept of Manifest Destiny also promoted the expansion of the United States as well which was a factor as well. Not long after the Mexican American War started and more land in that region was taken from Mexico. The reason Mexicans did not object to American presence in Texas at first was due to the buffer the American presence provided with European powers to the north and there were no signs of imminent American aggression. EDIT: Read below comments to see important corrections.

According to Edward E. Baptist's recent book, The Half Has Never Been Told: Slavery and the Making of American Capitalism, it was because of a drive to find new lands to expand the thriving new slave plantation economy that kicked in after the 1790s.

Southerners complained about the Spanish-held territories west of Louisiana and how they blocked the expansion of slave plantations. When Jackson made it known that the US would officially recognize Texas independence, "Observers believed this was the first step in uniting the fledgling slave owner's nation [i.e. Texas] to the much larger one to its east [i.e. Louisiana]." After the Mexican-American war, "entrepreneurs of the domestic slave trade jockeyed to send 'the tide of emigration . . . flowing rapidly to Texas,' as a North Carolina enslaver put it. In the next five years, the number of slaves in Texas would grow from 5,000 to about 13,000."

What is ironic, considering the acrimonious immigration debate in the US today about "illegal" Mexicans migrating to the US, is that Mexico had formally invited Americans to come and settle in sparsely populated Tejas.