Because the current is proportional to the voltage difference, and that voltage difference changes as the capacitor charges.
At the start, the voltage difference is maximal. It's pushing the current as hard as it can through the resistor. But when those charges arrive, the voltage difference gets a little bit smaller, so it pushes the current a little less hard. Then once those charges arrive, the voltage difference gets smaller still. On and on it goes... smaller voltage differences, smaller currents.
Does that help?