Trust the math
The correct answer is because the mathematics and computer simulations show that you win more (lose less) when you stand on 16 vs T instead of hitting at high enough positive counts.
But for a more intuitive "feel" for it, when you have 16 very few cards will improve your hand and most cards will cause you to bust. When the deck is rich in 10s, it is better to not take the chance of busting trying to improve your hand, but hope that the dealer must take a hit and bust.
Now with lower totals, like 14 and 15, more cards will not cause you to bust as compared to 16, and you probably want to get those cards because if the dealer has to hit, he won't bust with those cards either. Here it is better to try to improve your hand and chance busting, rather than passively waiting for the dealer to bust.
The situation you mention, where sometimes you stand on 16 and yet the count still says hit 15, is because there is 1 more card (the 6) that won't bust your 15, but will bust your 16. You are in the zone where the chance of busting your hand versus the dealer busting is such that busting your 16 is more likely so you stand, but due to the additional card that won't bust your 15, it is still better to hit the 15.
So at some point, there are high enough counts where it would be advantagous to stand on 14 or even 13, but the ratio of 10s remaining in the deck has to be so high to overcome all the additional cards that won't bust 14 and 13, this situation is so rare it is not worth worrying about.
Here is some other advice. The game you describe has very poor rules. You might as well get used to not playing all the hands. In order to beat such a game, you need to be very selective about what hands you play. You should only be playing hands when you have reached the pivot point. This method (backcounting) involves standing back and counting the cards and only entering the game (wonging in) and playing when the count shows a player advantage, and then exiting as soon as the count shows no advantage (wonging out). You will spend a lot of time standing around and watching and will probably only play 25% of the hands dealt, but those 25% will be profitable hands.
It takes discipline to use this method, so you have to decide do you want to beat the game and make money, or do you just want to have fun? The sooner you get used to the idea of only playing when you have an advantage, the sooner you will start taking the casinos money instead of them taking yours.