는 거예요 and 고 있어요.
The two phrases mean largely the same thing, but there is a difference which is hard to explain.
밥을 먹고 있어요 => I am having a meal. (plain sentence similar in meaning to 밥을 >먹는 중이에요)
밥을 먹고 있는 거예요 => "You know, I am having a meal" (the tone is somewhat like this)
-고 있어요 corresponds to "be -ing" in English (숙제를 하고 있어요: I am doing my homework).
-는 거예요 is similar to "It is that ...", as in "It's not that I won't but that I can't" (안 하겠다는 게 아니라 못한다는 거예요).
거예요 is a contraction of 것이에요 - 것(thing; the fact that ...) + 이다(is) + 어요
This structure is used much more widely in Korean than "It is that ..." in English. It presents or explains something as a general fact (돈은 있다가도 없는 것이다), puts an emphasis on a statement, etc. A similar -ㄹ 것이다 form makes a statement of will, prediction, or future tense (나는 꼭 성공할 거야 / 내년에는 그가 돌아롤 거야). But this pattern is so prevalent it's difficult to say all the usages are only to express these senses.
The reason it is so frequently used might have something to do with the sentence endings in Korean.
Unlike English which doesn't have special sentence endings, almost every (especially written) Korean sentence has to end in one of the ending forms. Even though there are quite a few of them available for you choose from, it might still pose a problem, especially in prose writing, of the sentence endings getting repetitive. The -(ㄴ/는/은) 것이다 form probably provides a relief in this regard, by creating a level of indirection through a new structure with a different nuance. But this is only my conjecture.
Looking at your examples, a use case of 밥을 먹고 있는 거예요 might be if you are eating and someone is wondering what you are doing or thought you were doing something else.
To tell him what is happening, you might say 밥을 먹고 있는 거예요 instead of 밥을 먹고 있어요 to explain it more clearly.
It's like "You know, I'm eating", "I'm just eating in case you were wondering", etc.
Similarly, 그래서 축하해 주러 가는 거예요 probably evokes a stronger explanatory connotation than it would without 거예요,
So this pattern might have something in common with such expressions as "It's just that", "It so happened that", "It came to pass that", "So it is that", etc, providing an added indirection with an explanatory register.
Thank you so much for that detailed description.
If I may ask one follow-up:
Does the '는 거예요' pattern have any overlap with the '잖아요' conjugation?
No. I don't think -거예요 and -잖아요 are closely related or interchangeable with each other.
-거예요/것이다 is very broadly used in all kinds of situations, while -잖아요/잖아 is a colloquial expressions with more limited usage, mostly for real life contexts. -잖아 is much like the tagged question in English ("It's a fine day, isn't it?"), so it's not suitable in formal contexts.
They can be used for similar effect sometimes, but it seems more of a coincidence than a deeper connection.
For example, 학생 때는 열심히 공부해야 되는 거야 = ... 공부해야 되는 거잖아.
이런 상황에서는 대책이 제한될 수밖에 없는 것이다 = ?? (잖아 can't be used in a serious context like this)
Ooops. I made a mistake with the first.
This is the change from the second sentence to the first:
밥을 먹고 있어요 --> 밥을 먹고 있다 + (으는 거예요) --> 밥을 먹고 있으는 거예요
I made the example up but I've seen a number of times some text which uses this '~는 거예요' pattern. I don't really understand why this is done. The sentence is changed into an adjective using ~는 and the sentence is completed with 거예요.
If I find another actual example of this in real text, I'll paste it here instead of trying to make my own example. Maybe that will help a lot.
EDIT: I am spamming my own thread here! But I found an example of what I mean. Take this text:
그래서 축하해 주러 가는 거예요
Why not make 가 the verb in the sentence? Why is the actual sentence structured like an adjective with 거예요 attached?