It didn't - not really. At least, no one knew the facts on the ground.

The Adams-Onis Treaty of 1819 set the boundary line between the United States and the Spanish colonies with reference to several rivers and meridians connecting them: the Red River (southern border of Oklahoma), north along the 100th meridian (the western border of non-panhandle Oklahoma), then following the Arkansas River to its headwaters, and then due north to the 42nd parallel (the southern border of Idaho).

That was the border as negotiated by Secretary of State John Quincy Adams and Ambassador Luis de Onís, neither of whom had more than a vague idea what any of this looked like on the ground, nor where either of these rivers were. It was generally believed, for instance, that the Arkansas River ran close to 42 North - when it's actually all south of 39 North. This was a common problem. At least, Adams and Onis can be credited with a border that can be drawn on a map. In contrast, the Treaty of Paris had drawn the original US-Canada border "on a due west course" from the Lake of the Woods to the Mississippi, when the river's source is well south of that lake.

When Texas gained independence, the area still hadn't been surveyed. Nor did Texas ever have a universally-agreed southern border. They claimed the Rio Grande, and in the Treaty of San Jacinto (forced on Santa Anna at the end of their revolution) had ordered the Mexican army to withdraw south of it. Mexico never honored that treaty, though, nor was Texan authority ever established much beyond the Nueces River.

However, when the United States had annexed Texas and defeated Mexico, they wrote in the Treaty of Guadalupe-Hidalgo (in 1848) that the Rio Grande was not only Texas' southern border but also its western border. That retroactively put much of New Mexico inside Texas, including the longstanding settlement of Santa Fe where Texan authority had never even been heard of.

The State of Texas didn't get to enjoy its new borders long, though. In 1850, they tried to send Texas Rangers into Santa Fe to enforce state authority - but President Taylor claimed it as part of New Mexico Territory and ordered the federal army sent to drive the Texans out. Fortunately, before any shot was fired, Taylor dropped dead. The new President Filmore defused the situation long enough for Texas to cede its northwestern lands to the federal government in exchange for cancellation of debts.

So, the Texan panhandle in Colorado and Wyoming came into being due to a combination of poor geographical knowledge and expansionist legal claims. It was only retroactively part of the Republic of Texas, and arguably part of the state for two years, during which no state authority was ever exercised there. And in the end, Texas was glad to give it up.

The Adams-Onís Treaty of 1819 set the boundary line between Spain and the US, a line accepted a few years later by the new Republic of Mexico—and later by the Republic of Texas.

When Texas desired to join the Union, it wished to do so as a slave state, which the Missouri Compromise forbade north of 36º30’. So Texas relinquished the area north of that line to the US. Four years later, as part of the Compromise of 1850, Texas also relinquished what is now the eastern half of New Mexico to the US.