Why is being a Jew transmitted through the mother's bloodline?

It has nothing to do with “bloodline” but yes, Jewish status is inherited automatically through your mother.

İn other words, if your mother is Jewish you are too. The thing is, this is not about blood lines per se. If your mother was a convert to Judaism, the child is born just as Jewish as if she was a direct descendant of King David.

Although Jewish status is inherited through your mother, tribal status (which is mostly forgotten and irrelevant these days) comes through your father, so you cannot be a Cohen or a Levi unless your father was (for example)..