- What is the point of such a question?
The point would be to test your use of English. In fact, some exams (such as FCE, CAE and CPE by Cambridge English) have an entire section of the exam, dedicated to questions like this one. Although these adverbs have similar definitions and are often listed as synonyms or near-synonyms in dictionaries and thesauri, they can't be used interchangeably. A native speaker of English, or a proficient user would know this by the ear, without thinking about it. So, the point of this question is to establish whether a language learner has reached that level.
- Why is entirely the best fit for the example sentence?
Absolutely, totally and utterly can in many cases be used interchangeably, meaning:
with no qualification, restriction, or limitation
Used to emphasize a strong or exaggerated statement
None whatsoever: she had absolutely no idea what he was talking about
In the example sentence there is no exaggeration, nor does the speaker want to say that the other person has no understanding of the significance of the meetings at all (no understanding whatsoever); they rather want to say that the understanding is not sufficient.*
Notice that in ODO, only the definition of entirely doesn't include the word "absolutely". Another word that could be used in your sentence is fully.
Totally can be used similarly to entirely, but only in informal use. Your sentence is set in a business context (it talks about important meetings) so formal register is required.
- How do you know to solve similar questions in the future?
It takes practice and exposure to language. If you need an instant solution, a good way to check is to search the corpora because the material that went through a publishing process is often edited, and therefore more reliable than what you can find by a regular internet search.
This Google Ngram shows that entirely is the best fit (most often used). It doesn't show that other options are incorrect, because Google books isn't exactly bullet proof, since it includes all sorts of books, including self-published and those from vanity publishers, which may have been subject to dubious (if any) editing.
*In such test questions there is little context and we can't know for sure, but there is a possibility that the speaker actually does want to say to the other person: "You are absolutely clueless about the significance of these meetings", but since the conversation is obviously taking place in a business setting, the speaker wants to be polite and professional. One can say that the sentence in your example is far milder than this one.