You have to add a background in a separate layer. Preferably below your foreground objects/illustration. Then match up the foreground objects colour and gradients to parts of the bg in order to make it normal and how you want it to look. May take some work.
Just an fyi, the transparency in illustrator graphics are really an illusion so you can have an idea of how it would look. But they have to be designed that way in order to be true vectors. In other words scalable to any size. If you were to create your illustration and or graphics in Photoshop which the graphics can be truly transparent, the graphics would only be pixel based and there fore not truly scalable. If you were to enlarge that in Photoshop, it would look blurred. The vector graphics on the other hand no matter how large you make your image or graphics it will not get blurry and or pixelated. I hope its a bit more understandably for you now?
I hope that helps. 🙂