Entschuldigen Sie means excuse me,
but Sie means in English you in English and me in German means mich or mir.
So why is it Entschuldigen Sie instead of Entschuldigen mich/mir?
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You need to distinguish between contextual translations and literal (word for word) translations.
The contextual translations looks at what a person of the other language would say in a certain context (using as many literal translations of words of the source as possible). In English, if for example you want to ask somebody a question you will say ‘excuse me.’ In the corresponding German situation, you would say ‘Entschuldigen Sie.’ Therefore, the contextual translation is correct.
Literally, you need to approach this differently; you need to take each word, identify its grammatical properties and then think what the direct translation is in the other language conserving the grammatical properties. Let’s do this with the English phrase first.
We have the words excuse and me. The first is a verb in imperative (second person) modus. The second is a direct object form of the pronoun I. The second is the first’s object which is grammatically required in this sentence. Translating this, excuse becomes a form of entschuldigen while me would become mich or mir — but since we know the verb, we can fix it to mich (jemanden entschuldigen). German now has two second person imperatives; one formal and one informal one. The informal one is easy, it just consists of the word entschuldige. Thus one literal translation of excuse me is:
The formal imperative is slightly more complicated since the formal Sie is, grammatically, a third-person form. Therefore, the third-person pronoun must be part of the sentence. The formal literal translation therefore becomes:
Entschuldigen Sie mich.
When doing the reverse, i.e. translating literally from German to English, we see entschuldigen and Sie. The first is a verb form of to excuse and could be one of many without context. The second could be either nominative or accusative of the formal second person pronoun — which is, as stated above, grammatically a third-person one. Along with the context information that this must be an imperative, we realise (see above) that Sie is grammatically necessary since only grammatical second-person imperatives can lose the pronoun. Thus, we need to take both words together and translate them as one:
You read correctly, the literal translation of entschuldigen Sie is simply excuse; no mention in made who or what is to be excused.
For grammatical reasons, entschuldigen mich is wrong; I noted above that the third-person imperative requires its subject. You can use the informal entschuldige mich which would be grammatically correct. And you also can extend the formal imperative with the object (i.e. who is to be excused) — which would make it entschuldigen Sie mich.
In German "Entschuldigen Sie" is the correct grammatical form. "Sie" is a substantive as it's the formal version of you. In German "Entschuldigen Sie" is a set expression like "excuse me" in English.
It is possible to say "Entschuldigen Sie mich bitte" which means please excuse me. This is used for example for the sentence: please excuse me at the office, I am sick. In German: Entschuldigen sie mich bitte im Office, ich bin krank.
If you use say "Excuse me, could you repeat that again" the German sentence is: "Entschuldigung, können Sie das bitte Wiederholen" or "Entschuldigen Sie bitte, könnten Sie das wiederholen". Here you could even leave "Sie" out.
"Entschuldigen Sie" just designates who should excuse something. In informal context it would be "Entschuldige" (the "Du" is not added).
You can say "Entschuldigen Sie mich" or short "Entschuldigen Sie" or "Entschuldigen Sie meine Kinder" oder "Entschuldigen Sie den Lärm".
"Entschuldigen mich" or "Entschuldigen mir" does not work, because verb and pronoun don't agree.
It works the same with other verbs: "Schauen Sie sich das an", "Nehmen Sie noch ein Stück Kuchen", "Fahren Sie bitte rechts ran".
It is incorrect to use "Entschuldigen mich":
"Entschuldigen Sie" is an imperative. When you're addressing the other person as Sie, the imperative is always followed by "Sie":
So the "Sie" in "Entschuldigen Sie" is just the subject of the imperative. When saying "Du" to the other person, the subject isn't explicitly stated, as in English:
Therefore, when you address the other person as "Du", you would be able to just say "Entschuldige" - Excuse me.
The me part of Excuse me is often left out in the German way of saying it, but there are circumstances where you can include it, which makes it "Entschuldigen Sie mich" (or, when saying "Du", "Entschuldige mich"): If you want someone to move aside, "Entschuldigen Sie mich" sounds better to me, but if you want to add an explicit request, just "Entschuldigen Sie" is better: "Entschuldigen Sie, ..."