For a more detailed look, check out my lecture on the subject.
In a nutshell: a supercontinent, Rodinia, broke up a hundred million years before the Cambrian Radiation, leading to oceans becoming saturated with chemicals due to increased erosion from the continents. At the same time, the climate was becoming warmer. Together, these had two effects.
There was a huge increase in oxygen levels to near-modern standards (from 10% present atmospheric levels to ~90+%).
This increase in oxygen allowed the biosynthesis of proper tissues, including such polymers as collagen. The oceans saturated with chemicals, including calcium carbonate, silicates and phosphates, as well as the warm climate, made biomineralisation easy.
This basically all led to a huge increase in the amount of morphologies that could be formed: these new animals could now form tissues allowing them to move, burrow, swim, etc. The appearance of biomineralisation allowed the evolution of weapons and defences and their coevolution – this ties into the ecological aspect: the Cambrian is when ecology, as we know it, first came about, and it’s all due to these developmental innovations.
And this happened relatively quickly, in the span of 30 million years (50 according to some authors, 20 according to others, but those are pedantic debates), which is why we call it an “explosion”, when in fact it was nothing more than a radiation of the Bilateria (all animals besides sponges and jellyfish). Because all the phyla that we know today (and some extinct ones too) all arose during this time, we think of it as an explosion in morphological diversity.