Correction:
fossils appear rather suddenly.
Without the use of rigid elements,
life forms are severely limited in size, shape and movement. It is a basic law in construction: you need a skeleton for support.
The main event in the Cambrian explosion is the development of the ability to use chalk, crystallized CaCO2, to form supporting structures within the body.
From a purely construction technical point of view, this is a major breakthrough. With a skeleton, the body can grow larger, more solid, and lighter. New kinds of structures, with great advantages, suddenly become technically feasible: jaws, teeth, spines, claws, shields, limbs, wings, lungs.
So it really isn't surprising to find an enormous diversification of life forms, an 'active exploration' of the new area of possibilities opened up by hard bone construction. Once the ability is there, it is easy, biotechnologically speaking, for bone structures to grow to different sizes, which allows a different overall body size and different shapes of body parts.
It's even less surprising to see these life forms suddenly appear in the fossil record: the hard, chalk-based parts of the body are the only parts that fossilize well. When remains of a creature are found, it is very rare to find traces of anything but the chalk part; for creatures that don't possess a chalk skeleton, it is very rare to find any remains at all.
So while the Cambrian explosion may be an explosion of life, at its core it's an explosion of chalk. (As a matter of fact, all chalk in nature consists of the remains of chalk-producing organisms.)
A case in point; the eye. The eye doesn't fossilize: it doesn't contain any hard parts. What we really find in these post-explosion fossils is not eyes, but eye sockets! Eye sockets of hard bone are new and clearly beneficial: eyes are important and worth being protected. But the sudden appearance of eye sockets doesn't mean that the eye itself didn't exist long before.
It should be mentioned that the development of multicellular life forms with chalk skeletons didn't happen overnight, and many intermediate forms can be found on earth today. We can think of a multicellular organism as a huge colony of single cells that stick together and coerce each other into specializing into specific functions. (Some of the mechanisms and principles of this coercion are already known to science.) This specialization could come into being within creatures that already form colonies, but with little or no specialization among the cells. Examples of such creatures living today are sponges and corals.
It's easy to see how the use of chalk within the body may have developed gradually. Cartilage is structurally similar to bone tissue, but instead of hard crystallized chalk, it contains collagens; it is viscous enough to provide some rigidity, but still flexible, and too soft to fossilize. Cartilage is an important building material in life forms. There is nothing to contradict the evidence that creatures with hard bone have developed from predecessors with skeletons of weaker tissue, creatures that already contain most of the same organs, but are too soft to fossilize.
Jellyfish, for instance, owe their body shape and size to this type of tissue - note that some jellyfish, like the Portuguese man-of-war, are actually colonies of jellyfish-like organisms, rather than single multicellular beings.
Another example: we, the vertebrates, have developed from primitive fish with skeletons entirely made out of cartilage; sharks are among this type of fish.
(A personal note. My primary response to rk2001's writeup is not nearly as reasonable as what you see above. I feel outrage and shame upon seeing such views exposed in public. What would you do if you met someone who claimed, right into your face, that airplanes don't exist? They are heavier than air, so they cannot possibly fly! It's all a matter of clever illusion! Let's see how the avionicists answer that! What do you do, get angry? Roll your eyes, write the person off as a loss to reason, and change the subject? Or do you patiently point out that evidence to the contrary is all around, that the actual principles that allow airplanes to fly are well known and start to explain them? I really don't know the answers to that question.
Hurray to the civilizing powers of E2.)
PS to rk2001; if you feel it is inappropriate or rude to describe how your writeup makes me feel, my apologies.
PPS to LoMeta: Thank you. Good to know that the scientific jury is still undecided on what explains this event. In any case, the above shouldn't be treated as gospel.