Okay, right. Where to start. I'm simplifying this, uh, a bit.
The record of life through time is the fossil record. The fossil record is things that are preserved in rock. Rock gets laid down in a number of ways and at varying speeds. Sometimes you'll get three inches depth of sediment over a million years. Sometimes you'll get thirty feet. Sometimes you'll get three hundred feet.
Rock also gets destroyed in a number of ways. The longer a given body of rock has been around, the more likely it is to get destroyed soon. There are very few places on the Earth where unaltered rock from, say, the Archaean, are preserved.
The period of time of interest to us is the base of the Cambrian, 542 million years ago (Ma). This is, I'm sure you can appreciate, quite some time ago, even to those of us who work with deep time (for reference, the dinosaurs died out 65 million years ago, human ancestors arose around about 4-3 million years ago ish (this is another one of those 'controversial' issues), etc etc).
Unaltered Cambrian rock (by 'unaltered' I mean not squeezed to hellandgone by more rock on top, not baked by heat and depth, and outcropping at the surface - i.e., rock we can get at) is not actually that common. Most of it has been metamorphosed, subducted or eroded by now. So, baseline for understanding here is that the history of life through time is a book, and the closer to the beginning of the book we get, the more pages are missing.
Now, let's get back to that thing about sediment getting deposited over time.
Evolution happens over time. How much time is debatable (see 'Dawkins vs Gould' by Kim Sterelny for a good synopsis of the two sides of the debate on this one), but for the purposes of right now, we'll just accept 'a long time' and move on. So any given population of animals or plants changes by very small increments (okay, okay, there are exceptions, I'm not going into them here, just work with me).
Not all animals are preserved in the fossil record. Off the top of my head, I think the preservation rate is somewhere between one and ten percent of any given species may enjoy the luxury of fossilisation. It's all dependant on the environment they die in and a few other things. The great majority of fossils that are found are of marine organisms, because the environment is better for preservation under the sea.
So, we have basically a roulette here. Not many specimens are preserved. The rock they're preserved in is becoming increasingly rare. The ones that are preserved are not necessarily the ones that show any kind of easy-to-see change.
And there's one more thing. Generally, the only bits of an animal that fossilise are the hard parts - shells, teeth, bones. But I bet you can think of a dozen things alive today that don't have any of those.
This is where the term 'Cambrian Explosion' comes in. At about 542Ma, suddenly in the rock record we find multicellular organisms with shells. It's literally, bang, there in the rock. Like an invisible line - below it, only microfossils, if that. Above it, shellfish (not quite molluscs, but basically shellfish).
So, the reason I say 'sudden', is because in all likelihood it was nothing like sudden. It's just that it appears sudden because the rock is not preserving all the evidence - it's like a strobe light. You put on a strobe light, and dance. Watch your friends dancing. You only see flashes of what they're doing, not all the bits in between. If one flash of the light they were not wearing a hat, and the next one they were, it'd look to you like the hat just magically appeared. Of course it didn't - they had to pick it up and put it on, but you didn't see that bit.
Um, I hope that explanation helped.