This is just an addition to the given answers, which have only pointed at one single subject of this science -> grammatical issues in the Quran and the implied linguistic meanings.
A major reason or goal of the science of I'rab al-Quran is or maybe to show a couple of things:
The grammatical and linguistic correctness of a reading as you may find
- one reading using passive form while an other is using an active
- One reading using a plural where one is using singular.
- Some readings are using some kind of linguistic specification for example: any word that ends with هم no matter what the logical grammatical rule should be has a dama over the ha' "هُم" or a kasra over it "هِم" and similar specific rulings (There's a similar case for singular too). In books on the matter the rule behind it would be explained and the linguistic sources would be analyzed.
- There are -at first sight- some grammatical Issues with some readings, so this science solves them and gives an explanation.
Note that I'rab includes the root "Arab" or "Arabic" and could be understood as make it Arabic or check it's Arabic background ...
Another point which is shown is the root of some Quranic words in the original Arabic language for example the word "صراط" was originally pronounced "سراط" and some tribes used to pronounce it more likely with a "zayn" so that it might sound like "زراط" the three of them are all correct and approved readings of the Quran, while the most prevalent is what we may call a deviation of the original fasih language!
On the other hand even if Imam a-Shafi'i said that the Quran is Arabic, we know a more accurate "expression" would be it is Arabic or adapted by Arabs, as there are many words in the Quran which Arabs of the time knew, but former Arabs didn't as they came from Persian, Hebrew etc.
So to conclude this science is not only a grammatical backup for the correctness of the Quran, as in reality the Arabic grammar -in form of grammatical rules- is younger than the Quran. As it needed a canonical or unified language to create a concept of a more or less unified grammar (there are different language schools of grammar the most prominent are those of Basra and Kufa) and this only came by the inauguration of the Mushaf al-Imam by 'Othamn ibn 'Affan (May Allah be pleased with him). But it is also checking the origins of the languages of the Arabic tribes, established non-Arabic words and expressions and roots of some Arabic words.
One of the most prominent Books on the matter is at-Tibyan fi 'Irab al-Quran التبيان في إعراب القرآن by Abulbaqa' al-'Okbary أبو البقاء العُكبَرِي. The autor also wrote a book about I'rab al-Hadith (which is one of three known books on this topic other authors are ibn Malik the linguist and imam as-Suyuti who used Musnad Ahmad as a basis for his 'irab called عقود الزبرجد على مسند الإمام أحمد في إعراب الحديث). Other authors are abu Ja'afar an-Nahhas (here an online link to his book) and abu Isahaaq az-Zajaj أبو إسحاق الزجاج (here an online link to his book Ma'any al-Qur'an wa I'rabuh) a prominent author on this topic is imam as-Suyuti جلال الدين السيوطي who entitled his book simply 'Irab al-Quran إعراب القرآن
And Allah knows best!