There are hints throughout the whole movie trying to explain this.
First of all as a former martial artist, I can say that when you have trained as long and as hard as he did (7 to 14 years old, every day, all day), you are familiar with a lot of styles and can adapt to a lot of situation or new styles that you see/learn with a lot of ease.
It is also said in the movie that "you are what came before you". His mother was
a mystical being from another dimension that could manipulate wind, and the rings with her martial art, as seen in the fight scene against the father.
And his father is
a thousand-year-old human who fought a lot of wars to acquire power.
Which let us believe that he already has some kind of acquaintance with both, the use of the rings and his mother's martial art. We can see it in the scene with his aunt where
after she shows him the basic stance and some moves of their martial arts, he picks them up very quickly and uses them against her, sending his aunt flying a couple of meters away.
This part is to be taken lightly as there is only one sentence referencing to this.
His sister states that she was never trained because her father did not want to train girls. But as she, herself, says "... I learned by watching them and became even better than them.".
We can assume that Shang-Chi has seen his father use the ten rings during his training and has some idea on how you can use them. One example of that is when
his father went to kill his wife's killers with the ten rings, accompanied by Shang-Chi who confirmed it was them.
We should note that his sister trained in secret, which is why she is that skilled, but we have never seen Shang-Chi training with the rings, this is why the last part is speculation.
In the end there are similar situations during the movie where you could say "he has never been trained to do that":
Fighting on the side of the skyscrapper and almost falling multiple times.
Fighting in a bus and using everything at his advantage.
To answer your question, we can say that he is a highly trained and skilled martial artist which is why it seems so natural to him.
As a side note, when Shang-Chi uses the rings, it feels more evolved than when his father does. His father has a very linear style, where he mainly uses the rings to empower his punches by sending one or multiple rings. The only time he "goes out" of this linear style is when he uses moves that could be compared to the way you use steel chain in Kung Fu (the sister's weapon). It seems to me that the fresh mind of Shang-Chi and his new martial art allows him to come up with better ideas like
Sending rings flying into the air to walk on them and lend attacks.
Than his father stuck with the same mindset for so long.
We can also see that in the end of the movie, Shang-Chi created his own style by taking techniques from both his parents, it's at this moment that he accepts who he is. As seen in the last fight
He sends the rings inside the mouth of the demon, and sends them to his thorax. When falling he uses his mother's martial arts to make the ring rotates inside the demon (his hands are open IIRC). And finishes it off by using his father's move (closing his hand and bringing his fist up)