This is a pretty misleading statement. Dying during or after surgery from any cause (perioperative mortality) is relatively unusual, and is most frequently related to the reason one is having surgery or the patient's medical problems (comorbidities). Basically, really sick people having surgery or anyone having really risky surgery is at much higher risk of perioperative mortality. Healthy people having low-risk surgery are at exceedingly low risk of death (approaching but not reaching zero).
As an example, someone brought to the OR bleeding out from multiple gunshot wounds to the chest might have a 99% chance of perioperative mortality no matter how healthy they are. Or a patient with severe congestive heart failure, end-stage renal disease, pulmonary hypertension, and poorly controlled diabetes might have a higher risk of perioperative mortality even when having lower risk surgery (breast surgery, knee scope, etc). On the other end of the spectrum, a relatively healthy active younger person having their clavicle fixed has a very low risk of perioperative mortality (<<<<1%).
My other qualm is assigning risk of death during/after surgery to anesthesia. While anesthesia itself can cause death (lost airway, anaphylaxis, malignant hyperthermia, local anesthetic systemic toxicity, etc), the risk of death BECAUSE OF ANESTHESIA is a small subset perioperative mortality. Most perioperative deaths can be attributed to hemorrhage, infection, blood clots and pre-existing medical comorbidities. Statistics for perioperative mortality attributable to anesthesia vary wildly depending on the studies, but even the highest estimates are about 1:7000 (with other studies showing as low as 1:200,000).
I am just a bit sensitive about this as I feel like I spend too much of my day trying to reassure patients after their surgeon tells them their biggest risk from surgery is the anesthesia.
Long story short, there are a number of factors to consider when deciding whether to proceed with surgery. Anesthetic risk is one of them, but (for most people) shouldn't really be the main one.