¿a qué día estamos hoy? vs ¿qué día es hoy?
 

At https://www.collinsdictionary.com/di...sh-english/hoy I came across the following example: ¿a qué día estamos hoy? what day is it today?
I would appreciate it if you could explain:

1. Why is it {estamos} and not {es} ?
2. What does {a} in the {a qué} mean compared with the phrase {¿qué día es hoy?}

The question you found in the reference dictionary is merely one way to ask, "What day is it today?"
«¿Qué día es hoy?» is the exact translation of the English question.

The translation the dictionary gives asks the same thing, but in another way. It's a shame they didn't provide its exact translation.
The exact translation would be, "What day are we at?" or "At what day are we?" Both of these are correct translations into English, but the latter will sound a little weird to most speakers of English.
Now, armed with these translations, let's use the latter to explain the rest of your questions.
First things first. Your second question asks what {a} means. It means 'at'.
«¿A qué día ...?» = At what day ...?
Next comes «estamos». You likely know that this means "are we" when forming a question. (In a sentence, it would mean 'we are'.)
The last word «hoy» means 'today', of course.

All should make perfect sense now.

Thank you, Rusty, for your detailed answer. I would be grateful if you could answer a couple of follow-up questions.
From the point of grammar,
1. The answer to "¿qué día es hoy?" would be, for example, "Hoy es dia 15" or "Hoy es lunes". "Hoy" can be omitted in which case the replies are "Es dia 15" or "Es lunes" or just "Dia 15" or "Lunes". Is "hoy" (adverb) the subject of the question?
By analogy, in the English phrase "Today is Monday" “today” is a subject as well. Is it possible to retain the sentence structure as in "It's Monday today" in Spanish? All I can think of is placing "hoy" at the end: "es lunes hoy".

It seems to me that the concept of itness is somehow contained in the verb, it is expressed by the verb (given a context). Am I wrong?

2. It seems to me that the answer to ¿a qué día estamos hoy? would be (Nosotros/nosotras) "estamos a lunes hoy". The subject is either "nosotros" or "nosotras" (in case of a group with mixed genders the nosotros prevails). Is it the standard way to answer this question?
Please correct me if I am wrong in my assumptions.

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You may simply say, "«A 25 de marzo» or «A lunes»." You may include the subject pronoun with the verb, but it would be rare, since the verb ending ('estamos') makes it clear.

Thank you very much. Expanding on my questions in so minute a detail is a huge help and means a lot to me.