How does auxin kill weeds?

It is by accelerating internal growth that synthetic auxin kills weeds.

Specifically, a plant's life-sustaining energy is prioritized among a number of critical processes, such as flowering, leafing, rooting and shooting. A synthetic auxin occupies a plant's meristematic cells, which coordinate overall plant growth rates. A plant that grows too fast will have leaves that are lush and stems that are tall before either plant part is capable of supporting wider surface areas. The synthetic auxins in the herbicide 2,4-D will so overwhelm and weaken a plant that the weed will die from life-sustaining processes not being supported in the proper order.