I just lean that a telescope can't increase brightness of extensive objects and that at best, brightness will be the same as with our naked eyes. I still don't fully understand the physic behind this but I'm wondering why we would then use a telescope to observe large faint objects if it does't increase brightness?

Let's talk about exit pupil of a telescope, the size of the image formed behind the eyepiece.

The brightest it can be is when the exit pupil matches your pupil diameter.

For the sake of simplicity, let's assume your eye's pupil is 5mm, and your scope is f/5.

A 5mm exit pupil will be with a 25mm eyepiece.

In an 8" f/5 scope, the magnification will be 40.6x for the brightest image.

In a 16" f/5 scope, the magnification will be 81.3x for the brightest image.

But the object will be 4X as large by area as in the 8" scope. Its light will cover 4X the area on your retina and be far more noticeable.

Let's talk about magnification.

In the 16" scope, at the same power, the brightness will be 4X as great because the exit pupil is twice as large (as long as the exit pupil is not larger than your pupil).

For instance, at 203.2x in the 8", the exit pupil will be 1mm (5mm eyepiece). But in the 16", at 203.2x, the exit pupil is 4x as large (twice as wide)(10mm eyepiece).

So the 16" is brighter at the same power, and the image is much larger at the same exit pupil.

Both yield greater visibility in the larger scope.

Let's talk stars, which are just points. In that case, the 16" concentrates 4X the light gathering power of the 8" in each little point, making every star 4X brighter in the 16".

That's about 1.5 magnitudes brighter, so the 16" will see stars about 1.5 magnitudes fainter than an 8".

So, the bigger scope is a WIN-WIN-WIN proposition: it sees fainter stars, at equal magnifications all objects are brighter, and at equal exit pupils all objects are larger.

Larger = more visible (has to do with our animal self-preservation from our past)

brighter = more visible.

And the larger scope would have the advantage in all those cases.

But wait, there's more:

Diffraction of the star point yields a finite size to the star images. This is a "spurious" size, since all are far enough away they should appear as points to the eye.

The bigger the scope, the less diffraction of the star image, i.e. the smaller the size of the spurious disc.

This means the big scope resolves closer spaced dots, which means it sees closer double stars, smaller details on planets and moon, and more details in deep sky objects because

of improved resolution.

So the bigger scope resolves better in addition to the other advantages previously discussed.

So why a larger aperture? Those are the reasons. And they are pretty profound ones.