Since DNA doesn't replicate perfectly, offspring don't perfectly match parents. Slightly bluer feathers on a Blue Jay, or longer wings on a dragonfly, bigger teeth on a tiger...
all kinds of stuff.
If the environment is benign, none of this matters and most of the variants get to reproduce at the same rate as the rest of the population. Plus the variations don't show much, they tend to jitter around a norm.
Taking the Blue Jays...some are bluer, some are greyer, some fly a bit faster, all sorts of stuff.
Environment changes:
more rain consistently for several years: equals more clouds so the greyer jays survive better than the bluer or the normal ones (who, being easier prey for the hawks help to fuel a population explosion among the hawks). The normal poulation begins to look a bit different, despite the fact that female jays prefer to mate with bluer males (things are never neat in biological systems)
Eventually the "normal" jay looks greyer and flies faster than earlier jays did. The rest have been turned into hawk dinner so their genes have mostly disappeared from the population.
Now change the environment a bit.
We now have a normal population of slightly greyer, faster flying jays.
Does it stop there?
No.
You see, the climate didn't change all over so, over the mountain, there are still slower flying, bluer jays. But, of course, there's still that mountain. Some jays fly over, some don't, so the edges of the populations are still pretty variable.
But the population centers are more stable. Gradually on the grey side, those females who will tolerate greyer males reproduce more successfully.
More grey males and more grey tolerating females. We're on our way to speciation.