I will divide my answer into several sections:
#Why colonize
The Spanish Empire in North America primarily consisted of a handful of Spanish officials and colonists overseeing and extracting tribute and labor from indigenous communities and micro-states. This was the basic model everywhere from the missions in Alta California to the cahob of the Yucatán - islands of Spanish sovereignty in a sea of Indian subjects.
Spanish suzerainty was relatively secure in a place such as Mexico City - the seat of the viceroyalty, populated by thousands of Spaniards and with a sizeable militia. That was not true along the northern frontier. Spanish rule was threatened not only by French and British traders and expeditionaries but also by independent Native American communities such as the Apaches, Comanches, and Kiowas. In the 1770s hundreds of Spanish subjects were killed by raiders, while Texas itself was manned by a grand total of 1 villa, 2 presidios, and 7 mission outposts (Calloway p. 377).
This situation would not change significantly over the next 50 years. If anything, the security situation worsened during the early 19th century. The French seized Luisiana from Spain and then sold it off to the United States in 1803. The United States also grew as a threat - successfully seizing West Florida in 1812 and East Florida in 1819 (Henderson p. 33 - 34). During Mexico's 11-year War of Independence, massive fighting in Texas reduced the Spanish-speaking population from 4000 in 1810 to only 2000 in 1820 (Henderson p. 37). This will become relevant because the Spanish government - rather than the Mexican government - was the first to permit American settlers.
#Why no Mexicans
Some Mexicans (and Native Americans from outside Mexico such as the Cherokee) did migrate to Texas, but in general there was several inhibiting factors.
Population: The whole of (what would become) the First Mexican Empire contained 6 million people, most of whom were Indian peasants with social and economic ties to closely-knit communities (Henderson p. 35).
Poverty: Texas did not have any get-rich-quick incentives such as silver mines or major cities. What it did have was lots of desert and steppe (Chavez p. 5).
Security: As noted above, Texas was the scene of armed conflict between various parties.
As a result, there were not many people who wanted to migrate to the distant, poor, dangerous northern frontier.
#Why Americans
The Spanish had originally banned American immigration into Texas entirely. However facing widespread rebellions and wars across the whole of their empire, the Spanish finally relented in 1821 and granted Moses Austin permission to settle in Texas (Chavez p. 5). The hope was that the American settlers - who, after all, were willingly becoming subjects of the Crown - would behave as good citizens and possibly stabilize the frontier. An additional hope was that after the American frontiersmen jumpstarted new settlements, they would even attract Mexicans to migrate north as well (Henderson p. 36).
Moses never made it to Texas, dying en route (Henderson p. 36) even as Spain lost control of the region. However Stephen Austin pressed on and obtained permission from Governor Martinez to settle later in 1821. Stephen also traveled to Mexico City to confirm the grant with the Emperor Iturbide. This became the basis for the colonization laws of 1823 and then 1824. Both laws welcomed foreigners in to colonize and provided tax exemptions for several years (Chavez p. 48). In 1825, the State of Coahuila-Texas passed its own colonization law which was even more generous. Mexico would have preferred Mexican settlers, as all three of these laws did give explicit first-choice to native-born Mexicans over foreigners. However, they were willing to invite American settlers and perhaps incorporate them into the nation.
Why did the Americans want to come? Moses Austin told the Spanish that he intended to farm sugarcane and cotton along the Río Brazos (Chavez p. 5). Americans established farms and plantations along various rivers of Texas. By 1830 Texas had 4000 Mexicans and over 20,000 Anglos (Wasserman p. 76). Since the Mexican population was at only 2000 ten years earlier, the colonization laws had attracted some Mexicans - however, Americans constituted nearly all of the settlers. Alarmed at the rapidly growing American influence, Mexico banned further immigration from the United States in 1830 (Wasserman p. 75). Thus, the general free-for-all colonization period lasted only about 6 years.
#Sources
Calloway, Colin. One Vast Winter Count - The Native American West before Lewis and Clark.
Chavez, Ernesto. The US War with Mexico: A Brief History with Documents.
Henderson, Timothy. A Glorious Defeat: Mexico and its War with the United States.
Wasserman, Mark. Everyday Life in Nineteenth Century Mexico: Men, Women, and War.