(I could say this easier in Swedish, but I think I'll stick to English in case others are interested.)

Firstly, proteins can probably be synthesized faster that way, as many mRNA molecules can be made from one gene, and many protein molecules can be made from one mRNA molecule. (I've seen pictures of ribosomes lined up along an mRNA like pearls on a string.)

Secondly, it provides more ways to control the rate of protein synthesis, by regulating the activity of the enzymes that break down mRNA for example, or by regulating the activity of both ribosomes and transcriptase.

So it might not be absolutely necessary but it appears to be advantageous, and those who have advantages beat those who don't have them by natural selection. Also, some researchers think that life started with RNA and that DNA and proteins came later, so there might be "historical reasons" too.

Another idea: In an eucaryotic cell the chromosomes are in the nucleus and the ribosomes aren't. If the ribosomes were to work directly on the DNA, all of the protein synthesis would have to occur in the nucleus, and maybe that would defeat much of the point in having a nucleus. I don't think this can be a cause for mRNA's existence though, since (as far as I know) all procaryotes have mRNA too.