I kind of imagine this as a ping-pong ball going up and down. The ping-pong ball always travels at the speed of light.

Now ping-pong ball is put on a ship, and that ship starts travelling. The ping-pong ball is still moving at the speed of light, but as the ship travels faster and faster the ping-pong ball is moving sideways. From inside the ship the ping-pong ball is going up and down (and time stays normal, but only from the perspective of the observer that's travelling with the ping-pong ball). From an outsiders perspective the ping-pong ball is going to move in a zig-zag movement. As the ship goes faster the zig and zags will be longer and longer, and so (since the ball always travels at the speed of light) it's going to travel up and down slower.

Now. This is applied to all reactions between particles. Everything is going to move slower and slower since their movement is capped at the speed of light and more and more of their movement is taken up by keeping up with the movements in realspace. Until, when the ship travels at the speed of light, time becomes infinitely slow.