As a form of identification.
Under present law the documentation you have to provide to get a California Driver License or state ID card (for non-driver) can only be possessed by legal residents. See
http://www.dmv.ca.gov/dl/dl_info.htm#BDLP for a complete list of what documentation is currently accepted. A person in the country illegally or a non-resident alien just visiting can't get any of those items legally, and therefore cannot get a driver's license.
I started a new job in February. Because I work for a subsidiary of a major Canadian defense conglomerate and deal with a lot of valuable data I had to pass a background check, take a drug test, sign the biggest non-disclosure agreement I've ever seen, and present proof that I am eligible to work legally in the USA. The company accepted my California Driver License as adequate proof of the latter. I brought my US Passport to show them, but they declined to see it. Under the proposed legislation a California Driver License would no longer be useful for that purpose.
There have been several heated discussions in the J/PS forum about this as well - Under the proposed bill it would become trivially easy for an illegal alien to buy a gun in California. Right now the lack of a driver's license prevents them from doing that.
I'd have no problem with a bill to give driver's license to illegal aliens and non-residents as long as it had some kind of marking that clearly distinguished it from licenses issued to legal residents. We could kill two birds with one stone by marking licenses issued to convicted felons in a similar manner, to help prevent them from buying guns.