since we observe galaxies and other steller objects in very tiny tiny scale, and how we can understand that the speed of expansion on universe is higher than the speed of light ? even using the powerful telescope we just see the lights from a galaxy not the object itself in real time, also the lights are red shifted as well. So what's the basis of proving univers's expansion rate is higher than of light speed?
You don't see anything in real time.
When you check your phone, you look a nanosecond back in time. When you look at the Sun,$\!^\dagger$ you look 8 minutes back in time. How do you know your phone and the Sun are still there? Because you have a model of how the world works, and that model tells you that, to the best of your knowledge, they won't just disappear.
Likewise, we have a model of how the Universe works, based on theories and observational evidence for these theories. This model predicts that the Universe expands in such a way that two objects recede from one another at a rate proportional to their distance (the so-called Hubble-Lemaître law). Hence, for large enough distances, this rate exceed the speed of light. This is described by general relativity, and is not in conflict with the speed of light being a fundamental "speed limit", since that is a prediction of special relativity.
In physics, we can never be 100% sure of anything, but we can have more or less faith in our models. If you're asking more specifically how can we trust the models, then that would become a longer description of the assumptions entering the hypothesis, the theoretical framework, the equations, the observational evidence, etc.
Actually, the fact that we look farther back in time, the more distant an object is, has helped us developing our models, since this allows us to observe the Universe not only today, but throughout its entire history. This is one of the coolest differences between astronomy and other sciences, I think.
The redshifting of light is also sometimes helpful. The earliest galaxies (i.e. the most distant ones) are often only visible due to a certain type of light called Lyman $\alpha$. This light is ultraviolet, and hence doesn't easily penetrate our atmosphere. But when observing distant galaxies, the light has been redshifted into the optical or even infrared, and so makes it possible to observe from ground, rather than form space-based telescopes (which is much more expensive).