The phrase "popular sovereignty undermined the Missouri Compromise" refers to the impact of the concept of popular sovereignty on the political agreements made by the Missouri Compromise of 1820. The Missouri Compromise was an effort to maintain the balance between free and slave states in the United States by admitting Missouri as a slave state and Maine as a free state, while also establishing a line (the 36°30' parallel) that determined the status of slavery in territories north and south of that line.
Popular sovereignty, the idea that the residents of a territory should determine for themselves whether to allow slavery, emerged as a significant approach to territorial expansion in the 1850s, particularly in the context of the Kansas-Nebraska Act of 1854. This act allowed the settlers of those territories to decide on the legality of slavery, effectively negating the boundary set by the Missouri Compromise.
The author means that by allowing settlers in new territories to decide on the issue of slavery, popular sovereignty created conflict and tension that the Missouri Compromise had sought to resolve. This led to violent confrontations, such as "Bleeding Kansas," and ultimately contributed to the deterioration of the political compromises that had previously managed the North-South divide over slavery. In this sense, popular sovereignty can be seen as a destabilizing force that moved the country closer to civil war.