How can it happen to find infinite bases in $\mathbb R^n$ if $\mathbb R^n$ does not admit more than $n$ linearly independent vectors? How can it happen to find infinite bases in $\mathbb R^n$ if $\mathbb R^n$ does not admit more than $n$ linearly independent vectors?
Also considered that each basis of $\mathbb R^n$ has the same number $n$ of vectors.

There are infinitely many finite bases for $\Bbb R^n$, simply because we can always replace one vector by a scalar multiple of itself. There are infinitely many scalars, so there are infinitely many bases.
Note that for $n=2$ a subspace is simply a straight line going through $(0,0)$. There are infinitely many of those simply because we can just turn the lines through infinitely many angles. Similarly for higher dimensions, you have lines, planes, and so on, all of which you can twist and shake around to get an infinite number of subspaces.
If however the underlying field was a finite field then indeed a finitely dimensional space could have only a finite number of subspaces and a finite number of bases. In fact the whole space was finite!
While a finite dimensional space [over $\Bbb R$] is an object which is "essentially finite" (because a lot of its structure is determined by the behavior over a finite set) it is still infinite as a set.
