Think of the minus as taking something away and addition as giving something back. If you take something away from the things that are going to be taken away then it is effectively being given back. This is what the double negative is representing

For example:

You have $10 and owe your friend $6 so now you have $4 left. Now, lets say that your friend decides that he'll forgive $2 from the debt. So, the debt is now $4 (6-2 =4).

You can now subtract $4 from $10 and have $6 left over but you can also do this as a single equation:

10 - (6 -2) = 6

The brackets are clarifying it but we don't really need them - you can write it as

10 - 6 - -2 = 6

To summarise: You have $10 (+10), you owe $6 (-6) minus the $2 (--2 = +2) you were forgiven so you have $6 left after you pay your friend back

You can also do this with physical objects as a practical demonstration.

Make a pile of blocks, take some away then take some out of the pile that is being taken away and give them back. The ones you take away are the single negative, the ones you give back again are the double negative