This is part of my breakdown of the book "I Don't Have Enough Faith to Be an Atheist." Related posts can be found by clicking here.
One example I really like to help explain the gradual change in organisms is the gradual change in language. Some example point out that languages such as Spanish, French, and Italian are all derivatives of Latin. But it's not like one day people were speaking Latin and then the next day they were speaking Spanish. No, it is a change that would have happened somewhat gradually.
Actually, take the English language of today, for example. There are essentially a couple* forms of English — British English and American English. Over in Britain, they spell some of their words differently. Like "theatre," "centre," "programme," or "tyre." Or, they will use different words than we do, like "flat" instead of "apartment." (Here is a link with a number of other examples.) I would imagine the transformation from Latin to Spanish would have been somewhat similar. Differences would have been more subtle at first, but would grow further and further apart with time.**
* I think Australia uses English similar to that in Britain, but it would not surprise me if they have some of their own unique differences as well. Heck, we have differences here amongst Americans. I.e, "pop" vs. "soda."
** On a somewhat related note, I suspect in older times where many people could neither read, write, nor communicate easily with those hundreds of miles away, changes could happen more quickly than they can today. Today, we can have local dialects, but those dialects can't really evolve into their own language because we have to be able to communicate with those from other dialects, so we need to keep the overall language mostly the same (and we have to be able to recognize where they differ). And we need to be able to understand documents written hundreds of years ago. In older times, this would not have been as true, so local dialects would essentially be the language and could change somewhat with each generation.