Biological Baggage (emphasis on the logical)
Some Pokemon, even the mammalian seeming ones, shoot beams of intense cold out of their mouths (Dewgong), lightning out of their cheeks (Pikachu), and have actual burning flames as a part of their bodies (Ponyta). Others appear to be composed of non biological matter (Muk, Porygon).
Given all the laws of biology and physics they seem to break, a little thing like changing form, which real animals already do, hardly seems worth quibbling about.
But let's assume the evolving creatures in your world do not have these unusual abilities, they are just normal animals that undergo vast physical changes.
As Fayth85 mentioned, insects undergo fairly advanced changes as they go from larva to pupa to adult.
She /he also mentioned birds. Look at this picture of the Canada Goose. It is not just a size difference. The chicks have a much shorter proportional neck and different color and texture of feathers.
Not mentioned however are frogs. The change from a purely aquatic tadpole, to a legged, amphibious animal is fair drastic.
But none of those are mammals. However Humans are. During gestation, before certain hormones take effect, all fetuses appear physically female. They are not, technically the still forming reproductive organs are neither. But in terms of body shape they are, and males undergo a moderate change. Then again, when Humans enter puberty, they go another set of physical changes; growing additional hair, change in voice for male, breasts for females. Which bring us to topic two.
Even within a given species, the male and the female can have dramatic differences. Male lions are larger, and have a prominent mane that female lack. The male platypus is also generally larger, and also has poisonous spurs on the feet that females lack. Peacocks have huge tail feathers females lack.
There are dozens of other examples one can easily find, but the point here is that difference between individuals in a species need not be limited to age.
You want a biological way for your creatures to undergo a sudden metamorphosis. Like an egg hatching or a butterfly leaving the cocoon. In which case I point you towards reptiles.
When a reptile outgrows its skin, the outer layer is shed, revealing a new layer, which is generally much more vibrant, and in some species even a different color (some change colors as they age or based on environment or diet).
Your not-Pokemon could do a similar thing, especially if the change was not dramatic (Pikachu to Raichu rather than Teddiursa to Ursaring). The creature changes on the inside at a slower rate, but once it is ready, casts off its outer layer of skin/fur to reveal the new, 'evolved' form.
Or you could just not go with the 'instant' change, and then you wouldn't even need an additional explanation.