In the context of the passage, "popular sovereignty undermining the Missouri Compromise" refers to the principle that the settlers of a territory should have the right to decide for themselves whether to allow slavery or not, rather than having this issue predetermined by legislation such as the Missouri Compromise.
The Missouri Compromise of 1820 established a geographical line (the thirty-sixth parallel) that divided free states and slave states, thereby trying to create a balance between the two factions. However, the introduction of popular sovereignty, particularly through the Kansas-Nebraska Act of 1854, suggested that this earlier arrangement no longer held. Instead, it allowed new territories to decide the slavery issue through a vote, which effectively opened the door for the possibility of slavery expanding into areas that would have previously been considered free under the Missouri Compromise.
This shift in policy not only complicated the previously established boundaries around slavery but also led to significant conflict, as seen in "Bleeding Kansas," where violent confrontations erupted between pro-slavery and anti-slavery settlers. Thus, popular sovereignty weakened the resolution that the Missouri Compromise had sought to achieve concerning the status of slavery in newly added territories.