think of voltage as 'pushing power'
Your supply as a certain voltage (ie. pushing power). When the capacitor is completely discharged, the power supply will have to resistance to its pushing power. But as time increases, the capacitor charges (develops some type of resistance to the pushing power) and begins to slow its charge down. At some point, both voltage of the power supply and capacitor would be relatively equal, and so its harder for the power supply to 'push' the capacitor.
Pretend you are getting bullied. A big bully is pushing you around, and you take fall and cry. Then you tell a buddy about it, and he comes to help you out, now there are 2 of you and one bully, but the bully is strong enough to knock both you down. And so you tell another friend and so on. At one point the bully is going to have a hard time pushing you down because you got your buddies to help out.
Thats just a super dumb example, but it makes the point clear. At least I think so.