Although we know that day which we are talking about, why don't we say "What the beautiful day!" instead of "What a beautiful day!" ?
In addition, a friend of mine explained the following, but in my mind it is not persuasive—it has not convinced me yet. I am confused.
Generally, we use 'a' for a generic thing, when many of them exist, and use 'the' when there is only one of them. This also depends on the context. Here, we've experienced many other days, and will most likely experience many more, so it's 'a' beautiful day. For another example, you might go to purchase 'a car' (since the place you're going has many), but might take 'the car' to work (if you only own one car yourself). In politics, you would have 'a Senator' (since there are 100, and two to each state), but 'the President' (since there is only one at a time).