8 ideas
Every species tends to reproduce at rates that would quickly overrun available resources, but populations stay roughly stable. This gap forces a constant competition for survival among individuals, which is the pressure that gives selection something to act upon.
Rudimentary organs, awkward structures, and odd geographic distributions make no sense if each species were optimally designed from scratch, but are explained as relics of ancestral forms and past migrations. Reading the flaws and leftovers of an organism reveals its inherited history rather than its present purpose.
Major biological change occurs through the accumulation of countless slight, successive modifications rather than sudden leaps. The incompleteness of the fossil record explains the apparent gaps, so missing transitional forms are a failure of preservation, not evidence against gradual change.
Fossils form only under rare conditions, most sediments are later eroded, and only a small fraction of Earth has been explored, so the geological record is a book with most pages torn out. The absence of fine transitional series is what we should expect from so fragmentary an archive, not evidence against gradual change.
Descendants of one species that become more different from each other can exploit more varied niches, so they face less competition and leave more offspring. Selection therefore tends to push lineages apart, which explains why one ancestral form branches into many distinct species rather than simply changing in a straight line.
All organisms are related as twigs on a single branching tree, with living species at the tips and extinct forms as dead branches and forks. Seen this way, the nested groups-within-groups of classification are not an arbitrary filing system but a genealogy recording real common ancestry.
Darwin bred pigeons himself and showed that breeds as different as the pouter, fantail and carrier, which an ornithologist would class as separate species, all descend from the wild rock pigeon. If breeders choosing tiny differences over a few centuries can produce such divergence, nature selecting over geological time can produce far more. Artificial selection is the visible, small-scale model for the invisible natural process.
More individuals are born than can survive, individuals vary, and much variation is heritable, so any variant that slightly improves an organism's odds of surviving and reproducing will tend to be preserved and passed on. Summed over vast numbers of generations, these tiny retained differences produce new adaptations and new species without any designer or goal.