One of Darwin’s major points is that a non-sterile offspring cannot be produced by anything other than two parents of the same species. With this, fact, it makes it clear that new species must arise from existing ones as slight and small changes over time (p. 484). This allows him to explain the idea of a common ancestor from which other things descended. He also argues against the lack of fossil record showing all of the link species is because of the geological record being incomplete. With this, he also says that there are many species discovered by such records that may be varieties, rather than entirely new species (p.486). Another example he shows is how humanity causes selection amongst animals as well by exposing them to different new conditions (p. 488-489).
I think the most convincing is probably about non-sterile offspring only coming from two parents of the same species. It just makes sense in general, since a living thing cannot simply appear out of thin air, without parents, and for two parents to produce proper offspring they must be genetically similar. With this, there cannot really be any other way for new species to come into existence, if not by coming from other species. That argument alone helps set up a good basis for the slow process of change by natural selection that eventually creates new species. Most of his arguments seem decently reasonable, though some of them may have been slightly inaccurate, overall they make sense with the information he had at the time.