The well known example that if you travel into space you'd gain let's say 5 years and people on earth 25 in the same time or so.
I just don't get it and I can't find any logic explanation.
For instance: Two twins who came to live exactly at the same moment in the year 2000 and both die on their 75th birthday at the same time. One travels into space, the other stays on earth. Earth-brother dies on earthyear 2075,space-brother dies in earthyear 3050 or so...
I know its Einstein's point but that just doesn't instantly make it correct to me.
The other effect is that time in a strong gravitational field runs slowly.
If you move away from a clock, time seems to slow down as your distance to the clock gets larger and the time between a change on the clock reaches you over a longer period. But if you carry a clock in your rocket it will just tick at the same pace as on earth (minus the gravitational impact, which is measured but why does gravity have an impact on time...?)
The total velocity of our <x, y, z, t> vector will always be equal to the speed of light constant, c. You can think of something that has no physical movements as moving forward in time at the speed of light. As x, y, or z increases the magnitude of t will decrease so that the speed of light constant is always achieved.
Why this link has to hold is more complex and I cannot explain it well, but hopefully this gives some insight into time slowing as velocity increases.
Have a look at the simple inference example here: https://en.m.wikipedia.org/wiki/Time_dilation
Time doesn't necessarily slow down the further away you get from a clock. If you and a clock are both stationary (ie you're in the same inertial frame), you will observe it ticking in "normal" time, albeit delayed due to the distance. If the clock is moving relative to you however, you will measure its ticks to be slightly slower.
You may be confusing general relativistic effects which are distance dependent (as gravity weakens the further away you get).
If you carry a clock in your rocket, you will (in the rocket) measure it to tick once a second. When you get back to Earth, you'll find that it's lagged behind a clock that was started at the same time but was left on Earth.
Maybe have a look at simple wiki too https://simple.m.wikipedia.org/wiki/Special_relativity though it doesn't actually derive the Lorentz transforms unfortunately.
Ignore the gravity bit for now, that's general relativity and it's more complicated to explain.
To get an intuitive idea of why this necessarily results in symmetrical time dilation, imagine two people walking along non-parallel paths at a constant rate on a 2D surface. From either person's point of view, the other person has a one-dimensional relative velocity, either towards or away from the observer, and that relative velocity depends on the angle between their paths. One-dimensional acceleration is just rotation in the 2D space. Now, what happens if you project one person's path onto the other person's 2-velocity? The projection will be shorter! And remember, the direction of your velocity is the direction of forward time from your perspective. So, from your perspective, the other person has traveled less distance along the time direction than you have, because some of their constant-velocity path was used up traveling in space instead. I.e., from your perspective, time has slowed down for them. But, projecting your path onto their velocity vector also results in a shorter path--so the effect is 100% symmetrical!
Now, this analogy fails in two ways because the real universe doesn't have any meta-time that you can use to observer where the other guy is "right now", and because spacetime rotations are hyperbolic rather than Euclidean, but those two sources of error happen to cancel out nicely and you get the correct result that moving objects appear to move through time slower.
Now imagine horizontal as space, and vertical as time. In this case a 2D spacetime, but we can't really visualize 4D.
The reason they always talk about space-time in relativity is that you can't separate the two. If you want to travel faster through time, you have to travel slower through space. If you want to travel faster through space, you have to travel slower through time. There's an invariant like the length of that rotating clock arm called the "spacetime interval" that remains constant under the transformations that you have to do to go from one observer's perspective to another observer's perspective.
Problem is its in 4D so it's hard to visualize. There is a mathematical framework that can explain all of the transformations leading to length-contraction and time-dilation as simple rotations in a 4D spacetime (3 space + 1 time). It requires a bit more math, but then unifies things in a conceptually simple way.
But maybe just remember: "If you go faster through space, you go slower through time" "If you go faster through time, you go slower through space"
Your maximum speed in space is the speed of light, at which others will observe you as having no time passing.
Your maximum speed through time is one second per second, at which others will observe you as being stationary relative to them. Look up Alex Fluornoy's youtube video lectures. I'll edit this and link the specific one here later, if I can find it.
I've gone decades without hearing it explained that clearly and simply. Thank you (sincerely).