That’s a new way of looking at the church. I always thought it was the Catholic Church with 23 rites. so the rites are different churches?

That doesn’t seem right.
Actually, that is right.

Catholic teaching uses the word “church” analogously; it has several meanings:
In one sense, there is simply the Catholic Church, comprising a union of twenty-three autonomous churches, and in which the fullness of Christ’s Church subsists.
A single diocese itself is, in some sense, its own church. Every diocese is therefore called a
particular church - remember when Pope Benedict declared that Protestant denominations are not, in the strict sense, “churches”? That’s because the sign of a true particular church is the headship of a valid bishop… so every Catholic diocese is its own (particular) church.
Autonomous churches or
sui iuris churches are hierarchically distinct.
The clearest example of this are the patriarchal eastern churches, like the Melkite Greek Catholic Church, the Maronite Syriac Church of Antioch, or the Chaldean Catholic Church. These churches effectively run themselves: each bishop runs his own diocese, and if there are any issues that affect the whole church, their bishops meet in synod to solve them. These patriarchal churches also have patriarchs who help run things if necessary. The current Melkite patriarch is, I believe, Gregory III Laham, Patriarch of Antioch; the Coptic Catholic Church has a Patriarch of Alexandria, etc.
When a patriarch dies, the bishops of that autonomous church elect a new one. The only thing Rome gets to require is be
notified of who the next patriarch is once they’re done. Unless some exceptional problem requires special intervention from above, the pope of Rome literally never does anything else in/for these eastern patriarchal churches.
All the twenty-two eastern Catholic churches (and most of them have less autonomy than the patriarchal ones, I admit), and the Latin Church, make up together the Catholic Church.
It’s important to realize that these are
churches and not merely separate
rites, because otherwise one fails to acknowledge that they are (at least in theory…) hierarchically distinct.
My own opinion - and the Low Petrine Orthodox will, of course, disagree with me on this - is that this system, and particularly the relationship between the pope and the patriarchal eastern churches, is actually a very patristic system, one that does justice to both the first millennium practice and the teachings of the Vatican Councils on papal authority.
Of course, uniatism - the process by which all but two or three of these churches entered the Catholic Communion - is
not the ideal and has been rejected, but the combination of hierarchical autonomy in the eastern churches with a full submission to universal papal authority is indeed the Catholic ideal.