Your questions (or complains) are encompassing human knowledge acquired over a century and may be more! Science has several collections of known facts, huge body of information and knowledge. You have to know them just like in math you learned 2+2=4, and the area of the rectangle is length into breadth, and the ratio of circumference of a circle to diameter is always $\pi$.
Of course, you cannot read and find the history of someone did an elemental analysis and established the formulae of two solid compounds obtained from dissolving silver metal in nitric acid and copper oxide in hydrochloric acid respectively. Someone must have done an elemental analysis of the solids to find out the ratio of the elements in $\ce{AgNO3}$ and $\ce{CuCl2}$. You are right that you cannot find beforehand the charge on $\ce{Ag}$ in $\ce{AgNO3}$ if you had prepared the compound from scratch 150 years ago.
When chemists started studying the fact that some salts dissolved in water conduct electricity, with further refinements and experiments they were able to find the sign and the magnitude of the charge on the ions. Someone has experimentally determined it to be $\ce{Ag^+}$ and $\ce{Cu^{2+}}$ in these two salts.
You see there is a big body of factual knowledge in chemistry acquired by millions of experiments. You need a base to learn how chemical formulae are written. Once you collect information, you will start to see trends and patterns. Once you have “memorized” the basic charges and signs of common elements, patterns of precipitation, the question is answerable in a single line, as given by Prof. EdV. It is a precipitation reaction and silver forms insoluble salts when chloride ion is present.