I know all about how the fossil record shows more human-like species coming about over time, and how modern testing proves we have genetic similarities with other animals.
All that says is we have similar genetic blueprints to animals in the past and present. How do we know these similarities are caused by having a common ancestor?
We don't know, and we never will. Science doesn't work that way. But evolution is the simplest hypothesis that is both falsifiable and consistent with lots of experimental data. Therefore it is the currently accepted scientific theory.
I would agree that genetic similarities between current species does not in itself suggest that they evolved from a common ancestor. But knowing the genome sequence of various species let's you do much more than simply measure pairwise similarity between species. The structure of genetic variation (losses, duplications, inversions, mutations ... ) also fits with evolutionary models. And crucially, genome sequences confirm predictions made by the evolutionary theory --- evolution is not just a post-hoc explanation of data, it has actually been tested many times over, and has never been falsified.
A good example is the sequencing of the human and chimpanzee genomes. We knew long before any genome was sequenced that humans have 23 chromosomes (23 pairs), while chimpanzees have 24. If the theory of evolution is correct and we do have a common ancestor, then that ancestor should have 24 chromosomes, like the chimps do, and it must be that in humans, one chromosome is a fusion of two ancestral chromosomes$^*$. This is a strong, testable prediction: if we don't find this arrangement of chromosomes, then evolution is proven wrong! In the 1980's, the sought fusion chromosome was found using high-resolution cytogenetic banding: it's chromosome 2. Modern genome sequencing later localized the precise fusion point on the human chromosome.
Also, the fossil record provides a timeline, not just a bunch of more or less similar organisms. In general, we don't find old fossils of modern-day animals (including ourselves), but we do find old fossils of species that no longer exist today. And when piecing together the findings, it looks like there is a sequence of gradual changes over time. That's a pretty good clue. What hypothesis might fit this data?
If you are suspicious about the theory of evolution --- which is fine, you should think critically about everything in science! --- then you must look for an alternative hypothesis which is falsifiable and fits the data. (Note that "god made everything" is not a falsifiable hypothesis, since it makes no testable predictions, and therefore not relevant to science.) If you come up with anything, let us know!
$^*$ As pointed out in comments, a priori one could imagine other rearrangements between the ancestor genome and the chimpanzee/human genomes that give the same result, but I believe this is the simplest hypothesis.
It is vanishingly unlikely that when DNA sequences which are thousands and thousands of bases long are identical between two creatures, that those sequences have no common origin.
That's like saying, "Your Honor, I didn't pirate that Justin Bieber track; I generated random 16 bit audio samples which just happen to match!"
That is less likely than the proposition that Melania Trump's speech wasn't plagiarized from Michelle Obama's, which already sits at a laugh-inducingly low probability level.
It's not a question of what we know, but of eliminating vanishingly improbable, time-wasting hypotheses.
Not only do we have the statistical argument that even a vast number of monkeys typing on typewriters are not going to reproduce the text of Macbeth, but there is the observable fact that DNA replicates. We can almost literally see identical pieces of DNA being reproduced from a common origin. The replication from a common origin hypothesis to explain sameness is overwhelmingly probable, whereas independent origin is vastly improbable.
It's not a question of how do we know, but rather what is the most plausible hypothesis from among those that are consistent with our observations.
Knowing is reserved for pure facts, like consistency between logical and mathematical propositions. We can know that some theorem in mathematics proceeds from some axioms. We can also "know" the content of an observation, in some sense. We cannot know whether an explanation for it is true. We can know when an explanation is false and we can gradually repair our understanding by identifying these failing situations and improving the explanations.
A final remark. Just because two creatures have some common DNA doesn't mean that those creatures got that DNA from a common ancestor at the time when they diverged. DNA can be introduced by vectors such as viruses. So that is to say, two creatures that diverged, say, a hundred million years ago could have some common DNA that they acquired more recently, like a million years ago. It is practically certain, though, that this same DNA itself has a common origin.
Unless you regard viruses as your ancestors, then some common strand of DNA is in fact not necessarily evidence of common ancestry per se.
I think one very fundamental fact that you are overlooking, which we pretty much take for granted, is that DNA is the mechanism of inheritance of traits.
What I mean is that my DNA is very similar to my parents' and siblings. It's also similar, but slightly less so, to my grandparents, my cousins, aunts and uncles, etc. The further you go from my known relatives, the less similar our DNA becomes.
This is also true for all living organisms. For everything from bacteria, to plants, to flies, we can look at their DNA, and that of their relatives, and the more closely related (in the familial sense, as in parents, siblings, etc., not in the evolutionary sense, although I'll speak to that in a moment), the more similar the DNA.
So that is demonstrably true for every living thing we can observe today. But what about the past? Well, if this theory of DNA and familial relationship is true, then we can predict that, for instance, my DNA is more similar to the DNA of present-day people in the countries where my ancestors came from than from other countries, even though we have no historical/genealogical evidence of relationship. And this bears out-- a commercial DNA ancestry test shows I share genes with the general populations of the countries my recent ancestors are documented to have immigrated from.
This idea that familial relationship are biological relationship should seem at once obvious but also surprisingly insightful. If I am related to my human cousins because of the mechanism of DNA inheritance and the sharing of a common ancestor (our common grandparents), then it also stands to reason that chimpanzees are literally our familial cousins, very distant, from a common ancestor a long long time ago. Biological relatedness is in fact the same thing as familial relatedness.
If you don't buy that, then you have to explain why it's true that all observable organisms today share familial relatedness through DNA, but for some reason, this isn't true going back into the far past. What changed in the past? When did this start happening?
The only reason we get tripped up in this line of reasoning is because chimpanzees look very different from humans, and we cannot mate. In the past, people thought this was because different species were fundamentally different types of beings. But biological insights, specifically the understanding of DNA, shows that we are all fundamentally the same type of creature, with reproductive incompatibilities developing over time. The tree of life is literally a family tree.
Also, from biochemical understanding, we know generally how long mutations take to arise. Combined with fossil evidence, we can create a pretty good model of how far back divergences arose.
There are some good answers here already but they mostly focus on whether similarities in the DNA sequence of organisms imply common ancestry. This misses something much more fundamental that is very strong evidence for this particular question: we all use essentially the same configuration of machinery to store genetic information and turn it into proteins.
"There is nothing Goldilocks about DNA and RNA," Holliger told Science. "There is no overwhelming functional imperative for genetic systems or biology to be based on these two nucleic acids."
All organisms on Earth use the same nucleobases (G, C, A, T, U), even though alternatives exist that can apparently replicate perfectly well.
All organisms on Earth use the same genetic code for protein synthesis (with a few slight variations) even though the matches between codons and amino acids are probably largely arbitrary.
