The question appears to be simple, but in fact it is not.

You ask why is it that a negative number times a negative number is positive.

Well, if I were to ask you what is this "times", what would you reply? Think of it.

No one says "I travel by vehicle". But instead, "I travel by car, by bike, by boat, etc." depending on circumstances.

Similarly, although we use one and the same word "multiplication" or "times" in different circumstances, the meaning of the word is not always the same. What I mean is that in every case the operation must somehow be defined.

First one defines "a times b" for natural numbers 1, 2, 3, ... as repeated addition. Then one adds "0 times any number". Next one defines negative numbers and ponders whether it's possible to define multiplication for negative numbers as well. In principle,
there are many ways to do that. The common definition tries to - how could I put this? - fall in line with the definition of multiplication for positive numbers. How? For example, look at the table below:

5*4 = 20
5*3 = 15
5*2 = 10
5*1 = 5
5*0 = 0
5*(-1) = ?

Look at the right sides. They all differ by 5. It is natural to expect that "?" stands for -5, right? Similarly,

(-5)*4 = -20
(-5)*3 = -15
(-5)*2 = -10
(-5)*1 = -5
(-5)*0 = -0
(-5)*(-1) = ?

The right hand sides grow by 5. It would be natural to expect that "?" stands for 5.

There are other arguments in the same vein. "Negative times negative is positive" is true by definition for which there are many good reasons. Above I just gave you one.

All the best,
Alexander Bogomolny