2002 Ford Explorer 2002 Explorer died suddenly. Remote sta

I drove my explorer throughout the day, and didn't have any issues. I parked the car for 10 minutes, and when I went to start it, it wouldn't start. There was a faint clicking sound, and the lights wouldn't turn on. When I put my foot on the brake, what few interior lights and the satellite radio turned off. We tried jumping the car, but it still didn't start, but the clicking noise got much louder. While jumping, the auto locks started locking and unlocking randomly. I noticed the tail lights and parking lights are extremely faint, but were flashing even when the key was removed. I have a viper aftermarket remote start, which under normal conditions will cause the car to turn off if you put your foot on the brake, and will make the tail lights/parking lights flash while the car is running under remote start. I'm unsure if the problem is simply a battery, or the remote starter, or if it's something more expensive like an alternator/computer issue. I'm completely unknowledgeable about cars, and I am wary of towing the car to the dealership with no idea of what the problem might be. We did clean the battery connection because there was quite a bit of corrosion there. That's about it so far. Thanks!