Is it true that languages with average vowel inventory size (5-6 Vs) are most widely scattered? How can this be explained?
I'm going to take a guess that what your professor meant by "widely scattered" is referring to the location of the articulation of the vowel in the mouth, and not geographically.
The reason for this is because it's easier to contrast vowels that are articulated far from each other than to contrast vowels that are articulated near each other.
Take, for example, a fairly common distribution for a vowel inventory of five:
Front Central Back Close i u Near-Close Mid-Close Mid e o Mid-Open Near-Open Mid-Open a
The locations of the articulations of these vowels are as far away from each other as they can possibly be, making it easier to distinguish the different vowel qualities from each other.
Compare the similarly-shaped chart below, which has five vowels very close to each other:
Front Central Back Close i u Near-Close ɪ ʊ Mid-Close ɘ Mid Mid-Open Near-Open Mid-Open
While it's technically possible to articulate all of these vowels, no language is going to contrast five vowels so close to each other, when half of the possible space in the mouth is left unused, because it's much harder to contrast /i/ with /ɪ/ than it would be to contrast /i/ with /e/.
However, once you have a large vowel inventory, as in English, it becomes more likely that a language will contrast vowels that are so similar to each other, because there are simply more vowels to contrast, as you can see below (in a convenient rather than accurate analysis).
Front Central Back Close i u Near-Close ɪ ʊ Mid-Close eɪ oʊ Mid ə Mid-Open ɛ ʌ ɔ Near-Open Mid-Open æ ɑ ɒ
Here there's no wonder why we have all of /i, ɪ, and eɪ/: There are simply more vowels to contrast.
Similarly, there aren't languages that look like this:
Front Central Back Close i y Near-Close Mid-Close Mid e ø Mid-Open Near-Open Mid-Open a ɶ
Because there's no reason to contrast front rounded vowels with front unrounded vowels when there are no back vowels at all.
The same is true with consonants. There aren't any languages that look like this:
Stops Dental t̼ Alveolar t Retroflex ʈ
Because it would be much easier to contrast consonants pronounced farther apart:
Stops Labial p Alveolar t Velar k
The best resource on this question is the WALS map, which calls into question the claim. Before calling it into question, we'd need to determine what the actual claim is, in particular what makes a distribution "wide" versus "narrow". Theoretically, I guess, we could take the land areas of the planet, straighten out any curves, impose a grid of the area, and then observe how many cells have at least one small, medium or large V-count language in them. I seriously doubt that this has ever been done.
WALS reports languages as points, not as areas, so notice that the "distribution" of English is a tiny spot in southern England, French is a tiny spot south of Paris, and so on. This is completely wrong, as a statement of the distribution of those languages. Ainu, on the other hand, is reported as a tiny spot in Hokkaido, Japan, which is reasonably correct. Additionally, Austronesian languages generally have a medium count of vowels, and Austronesian is extremely widely distributed (by boat): also, there are more Austronesian languages than any other type.
Also recall that the languages reported in WALS are not selected at random: they are selected based on availability of information. The Central Sudanic belt in Africa is well-represented because the author and colleagues have done extensive field work in that area. The southern zone, populated by Bantu languages, is sparsely represented for other reasons – not that there are few languages in the area (compare to the case of Eastern Siberia or the Sahara, where language density is low).
The map doesn't attempt to state current actual geographical distribution, it simply states a property of certain languages on a list, and incidentally tells you a little be about what "Natügu" is, rather than making you look it up. The point corresponding to English has to do with where we historically know the language was originally spoken, and some of that knowledge goes back millenia.
It is mainly a consequence of basic statistics: Languages with an average¹ vowel inventory are also the most frequently occurring languages, the distribution thins out very quickly towards the more extreme values. And the most frequent type of languages tends to be found "everywhere" or "geographically spread out" while more extreme types tend to form smaller local clusters.
P.S. Given the small value of the average, the statistical distribution is not yet a Gaussian, but rather skewed (more like a Poisson distribution), and there are more languages with a "large" inventory than languages with a "small" inventory, as seen on the already quoted WALS map.
¹In a strict numerical sense, 5–6 vowels are not an average value but close to the median value. The avarage calculated as an arithmetic mean is somewhat larger, at least for the WALS sample, because of the skewness of the distribution.