How can a formula or assertion always be true, given any interpretation?
"Interpretation" is being used in a technical way. It isn't talking about interpretations of passages in natural or poetic language in which text might have literal or figurative meanings. In logic, it's talking about the Frege-Russell truth-functional theory upon which most of modern formal logic is based, in which every proposition receives a truth-value restricted to T (true) or F (false). That truth assignment is what's referred to as its 'interpretation'.
(A or not-A) is an example of a tautology. Let's look at that.
'A' can be interpreted in the linguistic sense to mean any sentence we like, but in the logical sense "interpreting" it means that (whatever 'A' means) we are assigning 'A' a truth value of either T or F, so that 'A' can be either
T
F
'~A' (not-A) will have the opposite truth value from 'A', namely F or T. (That's just from how these things are defined, the 'not-' operator reverses the truth value.)
F (if A is T, then ~A is F)
T (if A is F, then ~A is T)
'or' is a logical connective that connects two propositions. In truth-functional logic it has its own truth value which is determined by the truth values of what it's connecting (that's why it's called truth-functional). 'Or' is T if either one (or both) of its components is T. It's only F if both of the things connected by the 'or' are F. (That's just from how 'or' is defined in formal logic.)
These definitions of logical terms don't always correspond to how the same words are used in everyday speech, which leads to problems like the 'paradoxes of material implication' in which the commonly accepted logical definition of the 'if-then' relation leads to some very counter-intuitive results. Contemporary logic is working on making formal logic more congruent with natural language (typically by adding new logical operators for things like time, belief or possibility). This is where many of the controversies among professional logicians arise.
Returning to our example tautology (A or ~A)...
We have these two alternatives depending on what truth value we give 'A'
1. A is T, so ~A is F, so (A or ~A) is T because A is T (and the definition of 'or')
2. A is F, so ~A is T, so (A or ~A) is T because ~A is T (and the definition of 'or')
The "interpretation" is our truth value assignment to the simplest components (A and ~A in this case). In the case of a tautology, the larger compound statement is always going to be T regardless of the interpretation, in other words, it will always be T regardless of what T or F values the simpler components receive.
The opposite of a tautology is a contradiction. Contradictions will always be F, regardless of how the component parts are interpreted (the T or F values they receive).
To illustrate that, we can do precisely what we did above with 'or', except with 'and' this time. 'And' is defined so that it can only be true if both of the things it connects are T. (One of them no longer suffices.)
So
1. A is T, so ~A is F, so (A and ~A) is F because ~A is F (and the definition of 'and')
2. A is F, so ~A is T, so (A and ~A) is F because A is F (and the definition of 'and')
This stuff illustrates something else: why formal logic is "formal".
If (A or ~A) is always T, regardless of whether we interpret 'A' as T or F, and even regardless of what 'A' actually stands for in real life language, then the fact that (A or ~A) is a tautology is revealed to be a purely formal property, arising from the
form of the expression and not its content.
The same thing is true of the contradiction (A and ~A). It doesn't matter what 'A' means or whether we interpret it as T or F, it's a contradiction simply on account of its form.