What I'm going to do is explain a way of looking at relativity that should give a feel for what is going on. Some physicists may cringe at what I am going to write, but I have no shame!
First, let's take a look at Special Relativity. Einstein realised that there was a speed that was constant for everyone, no matter what they are doing. That speed is the speed of light. The constant isn't defined by what light does, but it is the ultimate speed, and light goes at that speed in empty space, as does anything massless. One way to look at why this speed is a constant is to consider it as the universal speed, the speed at which everything moves. This makes no sense until we think of the speed as being in 4 dimensions, with time as one of those dimensions. When something appears to be standing still (relative to you) in space it will not seem to be slowed in time. The reason is that something standing still (relative to you) in space is moving at the ultimate speed, but only through time. If something is moving relative to you, you will see its time slowed down. The reason is that that thing is now moving through space as well as time, so some of its speed through time has been diverted to speed through space. What happens as things speed up and slow down is that they are merely changing direction in the four dimensions of space and time; they aren't actually changing speed at all.
If something moves at the ultimate speed through space, as light does, it has no speed through time at all, and so its clocks (if it has any) appear stopped.
Now, General Relativity. Imagine you are sitting alone in the dark, and you turn on a light. A sphere of light will rush away from you in all directions, and (obviously) at the speed of light. If you are weightless, floating in empty space, then that sphere of light will stay centred on you, as it speeds towards the stars. That sphere of light will always be the furthest anything could have possibly moved away from you, because nothing can outrun light.
Gravity bends space and time. If you are on the surface of a planet, in the dark, and you turn on a light, a sphere-like shape of light will rush away from you, but it won't remain centred on you. Gravity drags down even light. The centre of that globe of light will drift, just a tiny bit, down towards the centre of the planet. If you were close to a very, very massive object such as a neutron star or black hole then the shape of the globe of light coming from you would be very distorted indeed.
The bending of space and time does this: it's constantly telling things that they should be somewhere else. The gravity of the Earth is nagging at your atoms, insisting that they really should be lower down. The surface of the Earth pushes back up on your body and your house, car and furniture, to make sure you don't fall downwards, but that nagging continues which is why you feel gravity as a force.
Remember from Special Relativity that everything moves at the ultimate speed through space and time. What gravity does can be though of as constantly moving space, and it's that movement of space that is telling your atoms that they should move. Because your atoms are experiencing a movement of space, they can't be moving at the ultimate speed through time any more. So, gravity's effect on space results in things being seen slowed down from a point of view where gravity is less.
At the edge of a black hole (the event horizon), we can consider that space is moving towards the centre of the hole at the speed of light. This has two effects. Firstly, than nothing can experience being stationary there, and secondly, time dilation will be infinite - clocks will appear stopped.
Everything moves at the universal speed, and gravity makes space move - that's an easy way to get some idea of how Special and General Relativity work.