How to explain cardinals equation $\alpha\aleph_0=\alpha$ where $\alpha$ is the cardinals of an infinite set?

Question

*Here $\alpha$ is always the cardinals of an infinite set.
We can explain $\alpha+\aleph_0=\alpha$ as that the cardinals of an infinite set don't change when the set united a finite or countable set. But how to explain $\alpha\aleph_0=\alpha$? Is it that a countable union of infinite sets with the same cardinals has exactly the same cardinals as those infinite sets? This is kind of weird but all I can think about. Any help would be appreciated! Thanks a lot!