In many learning contexts, the aim is for a skill to be so deeply ingrained that you do the right thing without any conscious effort - e.g. a musician hits a key or a string with just the right amount of force, a language learner uses the right tense or ending without thinking, a native speaker writes a word with the correct spelling etc.
But before this is possible, there is usually a stage at which conscious effort is needed, and during that stage various temporary aids for "getting things right" are used. Declension tables are such an aid. Their advantages lie in the fact that when a list is memorized, all items on the list can be easily retrieved by reciting the whole list from beginning to end (or until the desired item is reached); they are thus a very useful backup strategy for whatever method you use in order to gradually "develop a feeling" for correct endings.
Another advantage of declension tables is that they give a systematic overview and enable comparison: in many languages, two cases may share the same ending in some but not all paradigms, or one ending may correspond to different cases in different paradigms and all sorts of other confusing phenomena may happen. By virtue of conciseness, tables may be helpful to sort out the mess.
In developing teaching materials, the author considers aspects such as the age of the learners, their native and acquired languages, the amount of grammar instruction that they obtained in their mother tongue, previous experience with language learning, intensity of the course, etc. For example, I noticed that materials for English speaking students usually introduce individual cases one by one (and often did so even at the end of the 19th century - at least in Latin textbooks). This is because the whole concept of an "accusative" or "dative" is new to such students and needs to be internalized (besides the fact that the only English words where cases influence the form are pronouns, there additionally seems to be surprisingly little grammar instruction in the English speaking world), and throwing a declension table at them would be of little help. On the other hand, such an approach may be rather frustrating for students who have had a good training in recognizing a number of cases and paradigms in their mother tongue.