Edge padding is the excess bleed after the edges of the UV island. If your bg is black, the black is going to bleed in.
You can tell there isn't padding because of the bleed and most likely the texture you have only colors to the very edge of the UV's - it's redundant I know but hear me out. If you are baking in 3dsMax, there is the option to add edge padding - but this is only for baking, it won't do it do an already made texture.
In photoshop, you can make a copy of your final diffuse or normal or what have you, and add a subtle gaussian blur followed by stacking the layers.
Go to http://www.cgtextures.com/
, then Tutorials, scroll all the way to the bottom, the tutorial is called "Alpha mask background trick". Although it is only talking about using it for something with an alpha mask, I generally do this on all my texture at the very end. In the last part, he explains a more complex version dealing with lots of color change and solving it. Feel free to check out some of the other tutorials, they are very helpful and I consider them invaluable to have learned them myself.