Why, if species have descended from other species by insensibly fine
gradations, do we not everywhere see innumerable transitional forms?
We don't necessarily see the transitional forms, but they are everywhere non the less. Invisible transitional forms might take the form of increased intelligence or increased resistance to disease and numerous other beneficial traits that are not readily apparent on superficial observation.
The human race itself is an example of a species which is full of graduations that are readily visible to the human eye such as different, eye, hair and skin colours, different adult heights, different hair texture, different morphologies etc, etc. Dogs possibly have even greater diversity for a single species.
A problem is how we define a species. One definition of separate species is that they are unable to cross breed, but this is not strictly adhered to and difficult in practice to determine for moral reasons. One example is a few years ago the forest elephant was declared a distinct species from the African elephant and therefore deserved conservation funding in its own right. They argued it was a distinct species because they are smaller on average and preferred living in the forest unlike the African elephant. This is silly because by the same criteria, bush pygmies would be a different species which of course they are not. I suspect the definition in the forest elephant case was for funding reasons.
Why is not all nature in confusion instead of the species being, as we
see them, well defined?
Some species appear to be well defined from a superficial visual observation, such as penguins. I suspect this is due to sexual selection pressures. The females probably have a gene that gives them a preference for a mate of standard penguin shape and size and this drives a kind of uniformity, but beneath the surface the invisible evolutionary changes are going on such as increased resistance to disease, increased fertility, better fish catching skills, better ability to absorb and process nutrients and energy from the fish they do catch, increased resistance to cold and so on.
In a large population of a species, there is a lot of mixing of the traits going on, both visible and invisible but because of the shared gene pool the population generally evolves in the same direction as an averaged whole. However if a portion of the population gets cuts off from the the larger population (eg separation of land masses a long time ago) then the two populations can diverge due to different selection pressures and sometimes due to a disproportionate prevalence of a particular gene in the founder population due to statistical chance.
See Wikipedia divergent evolution.