nitrates are basically fertalizer. Fish waste turns into it eventually, and your plants suck it up, helping them grow.
I'll try to make a very long story short.
In a heavily planted tank there are 3 parts needed. Light, co2 and nutrients. Just to have numbers to refer too, lets give all 3 parts a value of 10. The number means nothing at this point... we will just call it the ideal level. When you have all 3 parts equal, you will have strong and healthy growth of plants.
So if you have your light at 10, co2 at 10 and your nutrients at 10, the plants can use up everything.
If you have Lights and CO2 at 10, and your nutrients at 5, then your plants will only suck up 5 of the light and 5 of the co2 as well. With out the 3rd part, they have no use for the extra light and co2. Those extra bits of light and co2 then trigger algea growth. If anyone of the 3 parts isnt at the ideal, the plants stop growing. When you pull that third part to 10 again, the algea will slowly fade. This also happens if any of the 3 parts are too high.
Does that make sence?