If stage 4 breast cancer mets to the brain causing multiple lesions, and a lot of swelling, why would a neurosurgeon suggest surgery? to do what?

Risk of herniation: It may be that one or more of the lesions is in a location which is at high risk for pushing the brain stem against the base of the skull causing a more immediate risk of death. It also depends sometimes on the status of the disease elsewhere in the body- if there was no other active disease, clearing it from the brain would put the patient in remission, even if temporarily.

Mets grow: They can increase in size and cause pressure on the brain and if they lie in critical areas then the patient will have a neurological deficits. So to prolong life and the quality of it it may be necessary to undergo surgery.

Depending on: The number, size, location, symptoms, activity of the breast cancer elsewhere, and time from your original diagnosis unitl metastasis has appeared, there may be a role for surgery, "radiosurgery" , which is a form of focal radiotherapy, m or while brain radiotherapy, discuss this with a team - medical oncologist, radiation oncologist, and neurosurgeon.