First, try jump-starting your car with cables from another vehicle connected directly to the battery for a few minutes, then start it. If it starts that way, the problem is almost certainly a dead battery, and you'll need a new one or at least charge it completely.
Now, if that doesn't work or the car shuts off shortly after starting, you definitely need to check the alternator. Something similar happened to me with my car a few years ago: everything seemed to point to the battery, the terminals were clean just like in your case, but when I took it to the mechanic it turned out the alternator had stopped charging. The alternator is what keeps the battery charged while the engine is running, so if it fails, the battery drains quickly even if it's in good condition. With a multimeter you can check: if the battery reads between 12.6 and 12.8 volts when the car is off, it's fine; but if it's below 12 volts, it's dead. The important thing is that you also check the voltage when the engine is running - it should go up to around 13.5 to 14.5 volts. If it doesn't go up, the alternator isn't doing its job.
Another thing to rule out is that the battery cables, even though the terminals look clean, could be corroded inside or loose. Sometimes the corrosion is inside the cable itself and you can't see it from the outside. Still, most likely you should start by trying cables from another car, because that gives you a quick answer without spending anything.