Why are countable sets considered small? Why are countable sets considered as small? 

It is something that I never really understood. In fact, I would rather say they are big since they contain infinitely many elements. Any explanation would be welcome.
The term "small" is subjective. If you are comparing countably infinite sets to finite sets, then yes, they are big. But if you are comparing countably infinite sets to other infinite sets, then they are small. For example, the set of real numbers is uncountable. This is shown by Cantor's Diagonal Argument.
I've never really heard anyone call a countably infinite set "small".  I've only heard people call uncountable sets "bigger".  Which makes sense in the intuitive sense that now matter how you try to shove an uncountable set into a countable number of slots you'll always have an uncountable number that don't fit so it makes sense to assume that an uncountable set has "more" elements.

Of course all of this language is casual and based an intuition developed in a context that simply no longer applies.  It'd be best to avoid using terms like "bigger" or "more" or "small" or "large" unless you specifically make it clear that a set $A$ being "larger" than $B$ means that no injective function from $B$ to $A$ nor surjective function from $A$ to $B$ can possibly exist.

In that sense calling countable sets "small" does make perfect sense as among all the possible cardinalities of infinity, countable is the smallest.  The only sets that are smaller than a countable set would be finite sets.

Which I suppose sounds a bit like saying the only things slower than the speed of sound are, pffft, subsonic activity, and scoff they barely count.

Everything is relative.

....

There are times that I convince myself that "more" is not the issue but that "thicker" is more appropriate.  If I take a bag of integers and dump them onto a geometric plane or line or number line... whoo-ee... there sure are a lot of them all over the place like a bunch of specks of dust.  But they sure are thin.  I can poke my fingers between them and push them apart. But the points in the plane or line....  They are THICK.  So thick I can stand on a plane and not fall through.

(Ain't language fun?)