Why does the side of the mebrane of a neuron with a high concentration of positive ions become electrically negative?
Q. “Wouldn’t it make more sense to then say that the high concentration site (more positive ions) has more positive voltage?”

A. No.

I know little about electrophysiology, but looking at the graphics and explanation that precede the screen from which the sentence you quote is taken the statement is justified. Where are you going wrong? 

Your argument implies that the side with the high concentration should be positive because it has more positive (K+) ions. But it has (initially) an equal number of negative ions (Cl–) so it is initially electrically neutral. (The graphic indicates this by having no charge on the K — not a very satisfactory idea in my opinion.)

The important thing to note is the premise — that the membrane is permeable to K+ ions, but not to Cl– ions. So, even though there is a concentration gradient, only the K+ ions can move across the membrane (the graphic shows those moving with a +ve charge). As they do so each leaves its ‘partner’ Cl– ion behind, so a negative charge builds up at the high concentration side. At equilibrium there may be a high concentration of K+ ions at the ‘left’ of the diagram, but there is an even higher concentration of Cl– ions!

Good luck with your further reading of these pages! It looks as if it gets much more complex as it goes on.