Image source: http://www.robotmonstertoys.com/apw/Clips/Weather/thermometer.gif
The temperature one morning may start at some temperature, increase by 10 degrees during the day, but then decrease by 10 degrees by that night. This means there has been no total change in temperature for the day, because the temperature that morning and then later that night was the same. So +10 and -10 are additive inverses.
Image source: http://www.sciencenews.org/sn_arc98/6_6_98/runner.gif
If you jog 2 miles down the road, and then turn around and jog 2 miles back, you will end up where you started. So +2 and -2 are additive inverses.
Image source: http://www.mathsyear2000.org/numberland/63/coins.jpg
If you have some savings in the bank, and then you spend all of your savings, you will not have any money in the bank. So the "positive" amount of money in the bank and the "negative" amount of money you spent are additive inverses.