Worms

In the Vendian Period (620-540 million years ago), worm burrows become increasingly numerous. Indeed, recently, larvae of segmented worms have been found in the Doushantuo formation of China. Since just prior to the Vendian the only evidence for macroscopic life was that of the worms, if the various phyla were to evolve, it must have been from the worms.
 

Arthropods

Many of the Cambrian creatures appear to be arthropods, the large classification of animals that includes insects, spiders, crabs, lobsters, and so forth. We can speculate on how these creatures evolved from the much simpler forms in the Vendian/Ediacaran era.  

An animal similar to the lobopod, Aysheaia might represent a step in this transformation:

It appears to be a 'worm' with ‘legs’, which are lobopods. A lobopod consists of muscles surrounding a blood-filled cavity. Lobopods are soft and pliable but they accomplish the purpose of moving the animal along. Each lobopod has a couple of claws at its end and the front two have spikes which apparently serve the purpose of grasping prey.

Anomalocaris may also lie in the line leading to arthropods:

Anomalocaris shows its relation to the lobopodians by having lobopods for legs. In addition it has a tail fan and flaplike projections that perform several purposes: protection, propulsion, and respiration:

“Of these, perhaps  the last was the most important. Imagine that in due course the flaps seen in Anomalocaris were modified. The leading edge now formed an elongate bar and behind it the structure was transformed in a series of trailing filaments. This, of course, would be reminiscent of the gills that arise above the walking legs of many arthropods. Thus, if the lobopod limbs were transformed into jointed appendages and the flaps into gills, then one could envisage, at least in broad outline, the transformation between an animal similar to Anomalocaris and a fully fledged arthropod.” - Conway Morris (1998, p. 184)

Thus Anomalocaris has traits that link it to the lobopods as well as to the arthropods.

But it also has traits that link it to even more ancient ancestors. Its mouth is wormlike, with teeth radiating around the mouth itself. Being related to worms is not inconsistent with being related to arthropods IF the animal is in the process of evolving from worms into arthropods. The mouth of Anomalocaris is seen at:  http://senckenberg.uni-frankfurt.de/

A vast variety of arthropods and closely related animals evolved:

Wiwaxia	Opabinia
Halucenia	Trilobite

So broad is the variety of such animals that some have thought that they may have represented new families of animals that have since died out and have no modern surviving representatives (Gould). However, a careful analysis of basic features shows them to be part of the arthropod branch of animal designs (Conway Morris).

So, what we find in the fossil record are animals with all stages of the transition between worms and arthropods: 1. worms; 2. lobopods; 3. Anomalocaris with both lobopods and arthropod appendages and possible incipient arthropod gills; and 4. arthropods.

Brachiopods

Brachiopods are another distinct Cambrian animal form. The evolutionary lineage of the Brachiopods appears to begin with a creature called Halkieria evangelista from the Sirius Passet formation of Greenland which is about 525 million years ago.

This animal has three broad zones of hollow scales. These scales are hollow and are inserted into the animal's body in such a fashion as to make a chainmail-like armor covering the body of the metazoan. These small scales overlap and make a flexible armor with which to defend against attack. The back of the animal has palm-shaped scales, the sides have knife-shaped of scales and the edges of the belly have sickle-shaped scales. This will become important later when we talk about an animal transitional to the annelids. The bottom of Halkieria was soft, without scales and could best be described as being like a snail's bottom. It enabled the animal to glide across the ocean floor. This fact will become important when we talk about annelids and mollusks. Halkieria can be seen at http: http://www.palaeos.com/Invertebrates/Procoelomates/Halkieriida.html

The shell at the ends of Halkieria is very similar to the shells of the brachiopods seen in the earliest Cambrian strata. What is believed to have taken place is that the two shells at each end of Halkieria were used by this animal as protection devices. The animal would curl up between the plates when threatened. There are other facts that support this interpretation. The edges of brachiopod shells have chitinous bristles, called setae, which extend away from the shell. The microscopic structure of the setae is identical to that of the chaetae of polychaete worms and we will see that these evolved from the scales of Halkieriids. In some of the brachiopods the setae are segmented as is the case with Halkierian scales.  And finally, "Neocrania", a modern brachiopod begins life as mobile animal that moves across the ocean floor, bearing the setae of a brachiopod but lacking the shell. Eventually it settles down, folds itself in the middle, secretes a shell and remains fixed like others of their kind. This is the type of action Conway Morris postulates as the steps leading from Halkieria to brachiopods.

Why????
The most fundamental question about the Cambrian Explosion is why did it happen. There are many ideas, and of course it is possible that more than one of these effects worked together.

Material from http://home.entouch.net/dmd/cambevol.htm, G. Morton; "Wonderful Life," S. Gould; "The Crucible of Creation: The Burgess Shale and the Rise of Animals," by S. Conway Morris; and "Evolving Brains," by John Allman..