Its hard to describe the effect because that is the question, this kind of worn out, painted like, old effect. My guess is designers used some texture and used overlay, am I right?
So I tried to replicate it and hopefully this will help you get to that effect.
First you need your final artwork (which in this case I will use as a turquoise layer)
Then you need a rought texture like this one:
Then open photoshop and add it to a layer and desaturate it.
After this you will invert the colours of it so the white will be black and black will be white.
Add a new layer
this layer will have your artwork although in this case will be just a full colour layer
It is important that your artwork is behind the texture layer. After this you will select an Opacity mode on the layer box for your texture. Select Lighter Color. Obviously depending on what colour you have on the artwork there can be other Opacity filters that best suit you. Feel free to experiment because it won't take that long and you will be able to compare them and their results.
The result will be something like this. If it doesn't lool like the image below try using levels to increase or decrease the amout of light of your texture. It will make the effect stronger. (for levels keyboard shortcut try Ctrl+L or Cmd+L)
If you are not yet happy about the texture you got (and this may happen because we were pretty rought handling this texture pushing blacks to whites and all that) try to increase noise in your texture.