A grammatically complete sentence should be able to stand alone without a feeling of incompleteness, but it doesn't necessarily mean that a sentence can be taken out of context and understood perfectly.
In the sentence "I, too, pay taxes," you have a subject ("I") and a verb (to be more prescise, a predicator, "pay"). In addition, the predicator also takes an object ("taxes").The sentence is grammatically correct, and makes semantic sense as well (you know what it means).
Here is an example of a sentence which is grammatically perfectly OK, but you have no hope of understanding it:
She does that every month.
Who is "she" and what does she do every month? You have no way of knowing: however, you know perfectly well that whatever it is this person, she does it every month -- and that is the thought that particular sentence was intended to convey. It is grammatically correct as well.
Here are two examples of utterances which are not complete sentences:
Pete told her a.
Tail its dog wagged the.
The first sentence either has a word missing or a word too many: it ends with an indefinite article, and in English the indefinite article cannot stand alone -- it needs at least a noun. It is not a grammatically correct sentence. "Pete told her" is a complete sentence (even though you have no idea what he told her); "Pete told her a lie" is also a complete sentence.
The second sentence has all the necessary words, but they are not assembled according to the rules of English grammar -- it is therefore not a sentence that makes any sense, just a list of disconnected words.