Allow me a detailed answer on this one, as it is indeed a good question.
Before we start, let met address your statement in bold. Instead of trying to find out what "evolutionists" "believe", why don't you simply learn some evolutionary biology? Perhaps it is what you meant, but evolution theory is a theory of biology. You don't have to rely on what people believe to learn about it.
Having said that, let's start.
The central key of evolution is inheritable genetics/traits and mutation.
Let's leave sexual reproduction out of this for a second as it makes the tree of life more complex and it would only distract you from the mechanism I'm about to explain (although it applies just as well for breeding pairs, off course). So let's assume for simplicity that newborns have only 1 parent.
(warning: hypothetical numbers ahead, for the sake of example)
In that case, a newborn is an exact copy of his parents + some changes.
You are, indeed, 99.999% the same as your parent. That 0.001 % difference is unique to you. That represents your mutations. All newborns have mutations. This is called the mutation rate. In human species, I think that number is about 175 mutations per newborn. The vast majority of which is actually pretty much neutral in effect.
When you get a child, then the same thing happens. The child is 99.999% the same as you, with 0.001% difference due to its own mutations. It also inherited YOUR mutations. So it is 99.999% the same as you and 99.998% the same as your parent. And so it continues down.
Your kid will have a kid, he will have a kid, he will have a kid, etc.
All of them will be 99.999% the same as their direct parent.
After a long time, still being 99.999% the same as its direct parent, it will be perhaps only 97% the same as YOU.
What is the result? Well... that every single newborn is always of the same species as its direct parents. And it gradually, very gradually, changes through the generations. At some point, you would look back thousands of generations ago and find out that BIG changes have taken place. At the core, that descendent far into the future will still have YOUR DNA, which was inherited through the ages. But it will have become a new species.
So all off spring of Homo Sapiens will forever belong to the group of Homo Sapiens. Homo Sapiens will not turn into canines. Canines are on another branch of this family tree. Homo Sapiens will no doubt speciate further into subspecies. But they are forever "stuck" in the Homo genus. That's their ancestry. That's the DNA they've inherited through the ages. And it ain't going anywhere.
You've surely heared that Humans evolved from primates. This is correct. What is also correct is that humans are STILL primates. That will never change. We are also still mammals. And tetrapods. And vertebrates.
We share an ancestor with all primates. We share an older ancestor with all mammals. We share an older ancestor with all tetrapods. Etc.
Too difficult to comprehend? Let's try an easy analogy. I love this: languages.
As you probably (or hopefully) know: Italian, French and Spanish are called "Roman languages". They are called that because they all derived from Latin.
At one point, the ancestors of Italian, French and Spanish speaking people all spoke Latin. Let's called Latin the ancestral species.
Now, how did we get from Latin to those 3 other languages?
Consider this: at no point in history did a latin speaking mother give birth to a spanish speaking child. Instead, each child was always brought up in the SAME language as its parents. But generation after generation, the latin changed. Some words were altered, accents were altered, dialects formed, fonetics changed. All in gradual manner. Again, at no point did a generation speak a different language then the previous generation.
Yet, here we are: French, Italian and Spanish. All different languages with the same roots. All three are still Roman languages. Just like humans are still primates.
Feel free to ask additional questions if you feel like you need to.
I'll be more then happy to do my best to answer them (or point you to what might be able to answer them) if you are sincere in your request.
