I saw a video on youtube of a Jewish guy saying that no one can can be a Jew unless he is born Jew or his mother is a Jew.
2 Answers
The sentence in your question is a tautology. Someone whose mother is a Jew is born a Jew
The definition of a Jew is the first example of a recursive algorithm. That is the rule is
A person is Jewish if he has converted according to Jewish law.
A person is Jewish if his mother was Jewish according to this algorithm.
This recursive algorithm goes back to the Revelation at Sinai, when every member of the Bnai Yisrael converted to the worship of Hashem as a unit.
Before that the rules were different, but we do not need to go that far.
For example see Is it possible for a Gentile to convert to Judaism? and My great great grandmother was Jewish - am I? which discuss this.
The source for this is discussed at Question 10.11: What is the origin of Matrilineal Descent?
The Torah does not always state every law explicitly. In the case of Matrilineal Descent, the practice is derived from Deuteronomy 7: 4,
Because he will lead astray your son from before Me
Is it true that for one to become a Jew he has to be either born Jew or his mother is a Jew?
No, that is a rather false. I have no idea how this odd idea came about or how it spread. But it is not true.
Think about it why would a Jew have to convert? If he is Jewish, he wouldn't, because...he is already Jewish.
A convert is following in the footsteps of Ruth, a Moabite, a non Jew that became a Jew and foremother to King Duvid.
I saw a video on youtube of a Jewish guy saying that no one can can be a Jew unless he is born Jew or his mother is a Jew.
That is the same thing. Jewishness only passes through the mother. But again, this is incorrect because we accept converts.