I'll address your questions in the descending order. We're given an argument from premises P1, ..., Pk to the conclusion C:
(1) P1, P2, ..., Pk ⊢ C.
We know nothing about the truth values of the premises or the conclusion. The so-called "trick" for transforming (1) into a valid argument is to append what the professor calls a "suppressed premise" Pk+1 to the list of premies to obtain the following argument:
(2) P1, P2, ..., Pk, Pk+1 ⊢ C,
where Pk+1 is defined as follows:
Pk+1 =def (P1 ∧ ... ∧ Pk) → C.
Substituting that into (2), we get:
(2') P1, P2, ..., Pk, (P1 ∧ ... ∧ Pk) → C ⊢ C.
Thus far, I hope, we have addressed your second question regarding the placement of the "suppressed premise". Now, let's see why (2') is valid.
Claim. (2') is valid, regardless of the truth values of P1,...,Pk or C.
Proof. An argument of form P1, ..., Pn ⊢ C is valid if and only if all assignments of truth-values that make all of the premises true make C true. Therefore, to show that (2') is valid, it will suffice to show that all assignments that make [ P1, P2, ..., Pk, (P1 ∧ ... ∧ Pk) → C ] true, make C true. Consider an arbitrary model M s.t. M makes all of P1, P2, ..., (P1 ∧ ... ∧ Pk) → C true:
α) M ⊧ P1, P2, ..., (P1 ∧ ... ∧ Pk) → C.
Since it makes P1, P2, ..., Pk true, it makes (P1 ∧ ... ∧ Pk) true:
β) M ⊧ (P1 ∧ ... ∧ Pk).
From (α) we know that M makes (P1 ∧ ... ∧ Pk) → C true, from (β) that it makes (P1 ∧ ... ∧ Pk) true. These two facts together, via modus ponens, give us the following fact:
γ) M ⊧ C.
Since it's impossible for M to make C true and make the conclusion (which coincides with C) false, we know that M makes the conclusion true as well. We had posited M to be arbitrary, so we're now able to generalize and conclude that all assignments of truth-values to the premises plus the "suppressed premise" make the conclusion true, regardless of the truth-values of those premises or the conclusion. ■
The professor is nevertheless wrong when he says that "You could always make any argument valid just by adding...". It's true, as we've seen above, that it's possible to transform an arbitrary argument of the form described into one that is valid by that "trick". However, the "trick" shows not that the original argument (e.g. 1 above) is valid, but that the augmented argument (e.g. 2' above) is valid. I hope it's now clear why I call the so-called "trick" ""trick"" instead of "trick" (I'd be happy to explain how those quotes work in the comments, if you're interested).