Bottom line- they can't.
Buuuuuut, if you are willing to stretch the definition of "zombie" you can get to something like "28 Days Later", where you have LIVING people who act at a very base, primal level, thus fulfilling most of the criteria for a "zombie".
Convert over to anerobic respiration so they have quick bursts of energy, then a long dormant period (like alligators). Possibly even switching to ectothermic (cold blooded) metabolic activity to make them less temperature sensitive (very slow in cold weather, but faster in warm temps/indoors).
This is a SERIOUS rewriting of how humans work however, almost all of our enzymes and metabolic pathways utilize aerobic respiration for energy and need a narrow temp range for adequate function.
Neurologically you can essentially kill off most of the cognitive areas of the brain, leaving motor control and base impulses like eat and fight to mimic typical zombie behavior. Of course this also would include mating, which is usually glossed over in most zombie fiction. It also has trouble with the "zombies don't attack other zombies" trope, one would normally expect primitive thinking beings to attack each other. So you would need a pheromone or some other signaling method to prevent self on self aggression.
So what you have is a debased human, with little ability for higher level thinking or language, but retains most of the physical motor control and all of the territorial aggressive behavior humans are known for. They can't do much for very long, but can survive for long periods of little food/water due to a radically lower metabolic rate due to changing to anaerobic respiration and ectothermic physiology. Of course this means they are limited in range and the first winter would wipe most of them out. But a "realistic" zombie is a time limited thing anyway as they are far too complex and fragile an organism to not take care of itself (humans living with no regards to hygiene, self protection, or shelter would quickly die of infection) so a true zombie holocaust would probably only last a few months.