Why a negative times a negative is a positive
For example, if you have a problem which is 5•-3 which you are not sure how to solve, you might go into a thought experiment. You wonder what 5(3+-3) is and you already have an idea of what to do when subtracting or adding negative numbers. Therefore, you solve the problem and get 0. Also, you test out the distributive property on the problem by multiplying 5 on 3 and -3 and you calculate down to the equation 15 +(5•-3). You are not sure how to solve a problem which you have to multiply a postive and negative. However, you know that the answer you get when using the distributive property should be equal to the answer to the equation 5(3+-3). Therefore you can conclude that 15+(5•-3)=0 and you now know that 5•-3 is 15.
You have another problem which is -2 • -6. Next, you wonder what -2•(6+-6) is. Due to the prior problem, you know that the answer to the equation is 0. Once again, you use the distributive policy and get (-2 •6 ) + (-2 • -6). You calculate down to this equation -12+ (-2•-6). You know that the answer to this equation is 0 as it is the same as the answer to the equation -2•(6+-6). Therefore you conclude that -12+(-2•-6) =0 and you now know that -2•-6 is 12.
Now you know why a negative times a negative is a positive.
Source: Khan Academy Video