Can someone conceptually explain time dilation?

One result of special relativity is that the magnitude of all 4-velocity vectors $\vec{u}$ is the speed of light. Written with the (-,+,+,+) signature:

$$\vec{u}\cdot\vec{u} = -c^2$$

One way to think of this is that everything is always moving the speed of light in some direction.

When I stand still, I move the speed of light in the time direction. My clock advances as fast as possible. When I look at other observers in other moving frames their clocks all advance more slowly than mine.

Imagine someone moving close to the speed of light. Their clock seems to hardly advance at all relative to mine.

If I start running, my 4-velocity vector is still the speed of light long. Now it has some non-zero component pointing in the space direction to account for my motion. That means the time component of my 4-velocity must have shrunk. My clock does not advance as fast as it used to, relative to everyone else, who remains standing still.

Clocks don't measure time and tape measures don't measure distance. They both measure the metric along a curve. Just like how a tape measure depends on the path so does a clock.

Take 4d spacetime, look at the path of the clock in 4d and note that at some of the events, it ticks. So at one point the clock ticks. And how far along the path should the clock go before it ticks again is 100% the entire question.

And it is based on two things. How deep a gravity well it is in, and how much of the curve is there between the two events, the start point and the end point of the curve in 4d.

Now curves are longest when going in straight lines, that's just how geometry dictates lengths in a Lorentzian geometry. So you can break down the curve into pieces and replace each little piece with a straight line and you've over estimated the length (in Euclidean geometry you'd be underestimating the length). So now you just need the length of each piece. You get that from the metric. Why? You can imagine there is some real time and the clocks are mean and just don't tick that way. Instead they tick based on a metric. The metric literally tells them how to tick. They tick because they measure the metric rather than measuring time.

You can imagine that the clock has to go along a 4d path and that the space and time flow through it as it ends up at different events. And for each little bit it computes the metric of that little bit and keeps a running total and when it gets to a certain total it ticks.

All you have to do is accept that clocks literally measure $\mathrm ds=\sqrt{g_{ij}\mathrm dx^i\mathrm dx^j}$ for each little piece and add it to a running total and then tick when that running total crosses the cut off running total.

And everything else too. When something is supposed to happen at a certain rate instead of waiting a certain amount of time, you go along a curve in 4d spacetime, compute $\sqrt{g_{ij}\mathrm dx^i\mathrm dx^j}$ along the curve and instead of doing it y times a unit of time you do y times every unit of $\sqrt{g_{ij}\mathrm dx^i\mathrm dx^j}.$

I think it might be more instrctuive for you to show yourself what is going on. If you work through the following example you'll come out with a decent understanding of the phenomenon. Consider a light clock onboard a ship that is moving at velocity v relative to an observer. Te light clock works by boucing light vertically between two mirrors spaced one light second appart. Given the postulate upon wich all of special relative is based "the speed of light is the same in every inertial frame" You can work out that the stationary observer sees the light in the clock to travel a greater distance tan the one moving with the clock. Therefore because the speed of light has to be the same in both reference frames time must be slower in the movig frame. i.e.

distance = speed x time

distance is less in te moving frame, speed is the same so less time must have passed. Therefore time is moving slower in the moving frame, hence time dialation. If you do the maths for the example you will be able to derive an equation for it.