I just want to re-state what borofergie said in a way that might make it clearer.
Basically, sugar in your blood sticks to your red blood cells, and the higher your blood sugar levels, the more sugar will stick to the cells. The A1c test measures the amount of sugar sticking to your red blood cells, and it is weighted towards the most recent months rather than the most distant months. So if your diet was bad in the month before your test, your A1c will rise significantly, and your good diet in the previous 2-3 months won't be reflected as much as your bad diet in the final month.
For me, the word "stick" is what clarifies the situation.
A regular meter reading just reveals the amount of free-floating sugar in your blood at that moment (or 10-20 minutes ago, if you are testing on your arm).