If $F$ is valid then $F \cup \{res(C_1,C_2,A_i)\}$ is valid
Since you are only asking for a hint:

You have to prove that, for any interpretation I, if it satisfies all
the clauses in $F$, then it also satisfies the resolvant
$res(C_1,C_2,A_i)$, and hence it satisfies all the clauses in $F'$.

From that you can infer the result to be proved, with some universal quantification.

Note on terminology and notations:


I would rather say that, when $F$ is valid, every interpretation
satisfies it. It is $F$ that is to be satisfied, not the interpretation. Or more generally, it may also be a clause $C$, a literal $A_i$, or a set of clauses $F$ that may (have to) be satisfied.
$F$ is valid iff for any interpretation $I$, $I(F)=1$.
You should not say "for any interpretation $I(F)$", since as you noted yourself, $I(F)$ is a truth value. An interpretation in propositional logic is (usually) an assignment of truth values to the symbols used in the clauses, i.e. a mapping of the set of symbols to the set of truth values. It also implies interpreting logical connectives in the usual way, and sets of clauses as a conjunction of the clauses. Then, truth tables allow you to associate truth values to each construct, up to the whole set of clauses, which is what is meant by $I(F)$. The interpretation is $I$. while $I(F)$ is the truth value it gives to the set $F$ of clauses.


Interpretation may be defined in a more general way, but the above should be enough here.