Without the apostrophe ("menus") the pronunciation would be different: the "u" would be pronounced as a 'schwa' (like the English "uuhhh...."). It wouldn't sound like the plural of the word "menu" anymore. The "u" in both "menu" and "menu's" is like the German sound "ü" and the "u" in the French "tu" (you).
I found this in the tips of the website I honestly think it's more clear than the reply you got
NOUN + S If the noun ends with a single vowel, you cannot just add the -s. Instead, you first add an apostrophe, and then add the -s. For instance, menu becomes menu's. There is an exception to the above rule: words ending with an -e. Those never get the apostrophe.
It's not correct how the man pronounces it. The way the man pronounces the word it would be written as mènuis and that is not a Dutch word. I will try to write how you should say it. Something like 'menuus', so a long 'u' and not like 'ui' as in huis (house), ui (onion). The pronounciation of the 'e' is wrong too. The way he pronounces the 'e' it sounds like the sound of a sheep (meheheheheheh) and your mouth is opening horizontal. It should be opening like pronouncing an 'o'. So more in a round form.
In Dutch, all nouns in the plural form are preceded with the 'de' article, regardless of whether the noun is a 'de' or 'het' noun in its singular form. :)
- Het huis / De huizen = The house / The houses
- De deur / De deuren = The door / The doors
- Het kasteel / De kastelen = The castle/ The castles
Another article that might help you: http://www.dutchgrammar.com/en/?n=NounsAndArticles.11