I was wondering how time propagates through the spacetime and how its pace is slowed down by effect of gravity? Without understanding the propagation of time , I found it impossible to understand the slowing down of time? It would be much better and highly appreciated if it can be explained with an example.
Personally i feel like using the word "time" here is a bit confusing. I prefer to think of it as the rate at which systems evolve. Now i'm sure some people will disagree but the concept of time being defined by this evolution, is more of a philosophical question if you ask me. An analogy i've found helpful is thinking that the universe "updates" itself at the speed of light. Whether that has any grounding in reality, i'm not really sure.
So to answer your question, the faster something moves, the slower it physically evolves. For example, an atomic clock uses the transition of energy levels of the caesium atom as it's point of reference because for a particular inertial frame, the interval is always the same. Relativity tells us that the faster that atom moves through space, the slower that transition becomes. With that in mind, gravity accelerates mass. Acceleration changes inertial frames and with it changes the rate at which a system evolves.