how come texas peacefully agreed to give up being its own country and be a part of the united states?

Answer: The main reason for this was slavery. The US did not want to annex Texas because doing so would have upset the balance between slave states and free states that had been accomplished with the Missouri Compromise of 1820.

Citizens of the new republic were overwhelming in favor of Texas becoming a state in the United States-many motivated in part by concern that Mexico might try to re-establish its oppressive rule over Texas.