Ultraviolet light kills cells (including bacteria) by damaging their DNA beyond repair. UV, especially UVC, is in the wavelength range that is shorter than visible light and can penetrate into cells. Cells are generally pretty good at recognizing and repairing DNA damage. However, if repair is not achieved, this results in mutation, which can be deleterious, such as in the case of mutations that occur in genes that drive the onset of cancer - think UV exposure (in the UVA and UVB range) via sunlight. Under extreme circumstances, as in the case of UVC exposure, the DNA damage is much more extensive and often cannot be repaired. Thus, the cell's set of "instructions" for how to respond to stimuli, how to carry out metabolism, and generally function is compromised, leading to death.

In 1878, Downes and Blunt published a paper describing how shortwave light could kill ("sterilize") bacteria, so we have known about this for a while. A highly cited paper from 2012 described how using UV light to treat infected wounds in mice was effective - at least as good as if not better than standard antibiotic treatment:
T. Dai, B. Garcia, C.K. Murray, M.S. Vrahas, and M.R. Hamblin, 2012. UVC light prophylaxis for cutaneous wound infections in mice. Antim. Agents Chemother. 56:3841-3848.

This paper, coupled with the rise in antibiotic resistance, led to an increase in investigating the clinical use of UV in treating bacterial infections. Clinical use is tricky - wavelength, dose time and focused application are critical. Inappropriate use could lead to cancer in the patient's normal cells! (Take home message - avoid getting a sunburn!)

What is the nature (mechanism) of the damage that causes death? I'm guessing this might be the deeper context of your question - how can light damage DNA and thus kill a bacterial (or any) cell? This question is a great example of why we need to understand chemistry in order to fully understand biology.

UVC triggers a chemical reaction in the pyrimidine base pairs in DNA. UVC causes a photochemical reaction in the DNA. It disrupts the bonding between the complementary pairs and when the light is absorbed by thymines (one of the bases) that happen to sit adjacent to each other, it can result in a chemical reaction that causes covalent bonding to create a thymine dimer. Widespread disruption of the complementary base pairing and large numbers of unrepaired dimers can lead to catastrophic consequences for the cell.

Ultraviolet light is powerful enough to break chemical bonds. This includes the bonds that make up DNA. By irradiating bacteria with high-energy UV, you break up the DNA and thereby destroy their ability to make proteins, which are created using the information in DNA (the UV will break up the proteins themselves, too). Since bacteria live by making and using proteins in chemical reactions, destroying the DNA they use to make proteins and the proteins themselves that they use kills the bacteria.