Why do we perceive time?

Once you ask about perception, it is no longer physics question.

Anyway we do not percieve time. Look around and all you will see is present moment. The time is abstraction given by the memory of changes that were once happening. And to derive from this memory the flow of time is an abstraction. All the percieved memory is here and now, but there are also links that connects which events predated the next.

Take for example a diary. You know that one event was sooner than later because the later is written on the next page. From that you abstract away the flow of time.

Now to return to physics:

The time is defined by its measurement, and we choose such definition to make physical laws as simple as possible. So let us have some clocks, that have such a period of repeated events, that we are satisfied to call the time intervals between those events as same, f.e. pendulum.

to say something happened at some time, you will say the n-th click of the pendulum and the event happened at the same time. Then you write down to your diary (or better yet - lab report) at this click this happened. At the end of measurement you have huge diary with a lot of associations between clicks numbers and events. If the events are position (x) of some object, then you have graph $x(n)$. By defining time interval between two clicks f.e "1 second" you can get graph of position vs time $x(t)$ and then seek the laws that are consistent with the graph, f.e. $F_g=m a$, where $a$ is acceleration of the object, $F_g$ gravitational force and $m$ mass of the object.

In Newtonian physics saying that something happened at same time as click of some ideal clock is absolute. Einstein however realized, that this notion is absolute only if the two things happened not only at the same time, but also at the same place. Then you can say they happened at the same time, because you have seen them both happened at once. If they are not at same place, you must adjust for light speed and here all of the magic of special relativity comes in.

So time in physics appears only as a set of data together with their relations to clicks numbers. But the data are always percieved whole at any moment, you never see any past or future or flow of time anywhere.

You are able to detect a change in things around you — that an egg can break, the sun can rise, a colour can fade. So, you can detect different states of things. And those states are obviously not all present simultaneously. What should we call this "phenomenon", this "feature" of the world? We call it time.

The fact that everything doesn't happen at once, is given the name: time.

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So, it is not obvious when you claim that "everything [...] is travelling through the time in the same direction and speed". Because time is only noticeable when things change. Hypothetically, if everything was slowed down equally much - all events and behaviour, all chemical reactions - then we wouldn't notice. Because what we measure time from - the changes in our surroundings - would have been slowed down just as much as our perception.

You seem to be aiming towards the conclusion that we shouldn't be able to notice anything changing since we all move equally fast through time. But this idea is an upside-down way of thinking, because if we really couldn't detect changes, then we wouldn't have had a need for a concept like time in the first place.