One of the effects of traveling at high speeds is slowing down of clocks. I can understand gravity time dilation effect but not how would velocity affect clock speeds.
How correct it is, if I say that higher the speed, more object travels through space dimension therefore less it travels through time dimension; therefore slowing down of time?
Imagine, for example, that two different inertial observers, one sitting on a train moving through a station with uniform velocity $v$ with respect to ground. The experiment will consist of turning on a flashlight aimed at a mirror directly above on the ceiling and measuring the time it takes the light to travel up and be reflected back on its starting point. The observer sitting on the train see that the light ray follow a strictly vertical path (as fig. a)from $A$ to $B$ to $C$ and the ground observer see that the light ray travel from $A$ to $B$ to $C$ as fig. b. Both the observer see the light back to the starting point but it is clear from the two fig. that they traveling different distances. If ground observer think that the time taken by the light for the experiment is $\Delta t$ and the train observer claim that the time taken is $\Delta t'$ then
$\Delta t' ={\frac{2BC}{c}} $ and $\Delta t ={\frac{AB+BC}{c}}$.
And hence it can be proved that ${\frac{\Delta t'}{\Delta t}}={\sqrt{1-{\frac{v^2}{c^2}}}}$ This indicates that the train passenger's clock runs slow but it seems to the ground observer. Actually both observer's clock moving at the same rate.
A couple of things first...
1) Time dilation is not a consequence of high speeds, but of ANY speed - it just the effects grow large rapidly within about 10% the speed of light. Low speeds can have measurable consequences as is the case with magnetic fields for example.
2) All identical clocks "tick" away at the same rate under all circumstances,** metering out their proper-time distances, so be careful using expressions like "time slowing down" as if "time" were a thing itself that moved or flowed. This will help keep in mind that "time" is component, like space, used to locate a point in spacetime. It's a subtle distinction but one worth bearing in mind.
**Clocks in relative motion and various points in a gravitational field move along worldlines that are not necessarily parallel and thus project different time intervals (coordinate times) onto the other clock's worldlines. This effect is time dilation.
That said, your "How correct is" part of your question is not too far off. That is, the greater the motion of some other object through your space (greater spatial component) the less the other object moves through your time (smaller temporal component) because the spacetime interval ($ds^2 = dt^2 - dx^2$) is an invariant quantity - is what I think you meant to say.
If you haven't done so already I recommend reading Sofia's comment as she works through the arithmetic and this may aid your understanding.
