The scoring is done so that if you have 70% of your answers right, then you get the best average score by guessing 70%, not 60%. The increased penalty you get for getting 30% of those answers wrong is smaller than the increased gain for getting 70% of them right.

But that's true only as long as you really get 70% of them right; so changing your answer e.g. to 80% while being only 70% correct would decrease the average score, because then the increased penalty for getting 30% of those answers wrong would be greater than the increased gain for getting the 70% right.

Without understanding the log bits, you can easily verify this in a spreadsheet calculator. Make a formula saying how many points you get if you report probability R and if you really get P answers right. Playing with numbers, you will find out that for a given P, you get the highest average score for R = P.