i spoke to this issue recently in another question. i had not (until now) seen this question in the wings.
first of all, "fine-tuned fundamental constants" are dimensionless. the speed of light, c (which is really the speed of every other interaction considered to be "instantaneous", not just E&M), Planck's constant, h, the Gravitational constant, G, and the Coulomb constant, k, (or, alternatively, the Permittivity of Free Space, \epsilon_0), are not fundamental constants of the universe to be fine-tuned. that are really just anthropometric expressions of units that we humans have cooked up. Nature doesn't give a rat's ass what units human beings use to measure or describe things. they can all be set to 1, by choice of units and they are in Planck units.
e.g. all the speed of light needs to be is real, finite, and positive, and, if the dimensionless constants remain the same, c will continue to appear to we mortals as the same 299792458 meters per second. if "God" or some other non-mortal being changed it to (in God's POV) to 100 meters per second, we would not know the difference if all dimensionless fundamental constants remained the same. our meter would scale (in God's POV) and our second would scale in such a way that we would continue to observe light moving at 299792458 of our new meters during the time elapsed of one of our new seconds.
that said, there are a few dimensioneless physical constants, probably the most well known is the fine-structure constant which can be thought of as the square of the elementary charge when measured in Planck units. (i.e. Planck units are able to define a natural measure of electric charge without any reference to electrons or other particles. so then it's reasonable to ask "what is the elementary charge in Planck units and the answer is around 1/(11.7), a dimensionless number.) John Baez has a page that itemizes 25 fundamental constants for the Standard Model and 1 more for the Cosmological constant. none of these constants, so far, can be derived from any of the others nor any existing mathematical source (though numerologists have tried) and must be measured from nature. as more new phenomena is discovered (like dark matter or dark energy) and properties of the new phenomena are measured, new fundamental constants will appear on the list. as theoretical physics research makes progress, some of these constants will disappear from the list as they will be explained or derived from other sources. it is not true that if just any of these constants varied just a weee little bit, that the Universe would turn out so radically different that life, as we know it, would not exist. most of these constants, we have no idea how things would be different if some of those constants were different.
but certainly some constants are critical. if the masses of particles (relative to the Planck mass, to make it dimensionless) were significantly different (and i mean orders-of-magnitude different), it's unlikely that the "triple-alpha process" that cooked up carbon in super novae would have serendipitously had energy levels lined up just so that carbon would be abundant and so that the energy levels lined up a bit less well with oxygen so that not all of the carbon got pushed up into oxygen. that is a legitimate Anthropic coincidence that legitimately asks the question "how could that have happened just by chance", suggesting the possibility of purpose.
(remainder copied from other answer)
doubters or deniers of the remarkability of fine-tuning respond with the Anthropic Principle as a non-divine explanation.
i might point out that the Strong Anthropic Principle (SAP) requires quite a leap of faith itself and they will respond with the Weak Anthropic Principle (WAP) which is essentially a tautology or truism. (the SAP says that if it's possible for life to emerge in the Universe and for emergent life to evolve into sentient and sapient beings, then the Universe is compelled somehow for that to eventually happen. the WAP says essentially that "conditions that are observed in the Universe must allow the observer to exist.")
the WAP has to be true, because it's a tautology. and, because it's a tautology, it doesn't say much. it's like saying 5=5. a kinda "empty truth" (and i know i am semantically misusing that term, but i don't care; i am trying to make a point.) then they will say, "Well, the WAP provides a perfectly natural explanation for Fine-tuning by showing that the human perception of Fine-tuning is simply an example of selection bias.
then i will say, "For selection bias to suffice in removing any notion of the remarkability of Fine-tuning of the Universe, that requires a large collection of universes from which to select a life-friendly universe." then they will say "Yeah! Q.E.D!"
they may even point to different mathematical models of the multiverse, some of which are logically consistent and may even be true. but we will never know. because, for the same reason i will never create a God-measuring-device or a falsifiable physical experiment that will show, one way or 'nother, that God exists (or not), neither will they ever create a similar falsifiable experiment demonstrating the existence of any universes outside of the observable universe.
so it's fine to believe in the Multiverse. it might even be true (and even if true, that would only push the argument of infinite regress back a level; i would argue that God created the Multiverse). but i don't need the existence of the multiverse to have an understanding of the existence of me.
however, what if this forever unmeasureable Multiverse doesn't exist? what if the Universe we live in is the only one? that appeared some 13.8 billion years ago (perhaps as a hell-of-a quantum fluctuation)? well, for the 26 or so fundamental physical constants to come out just right so that sufficient carbon was cooked up but for the same process to not fit so well in cooking up oxygen that all of the carbon would have been converted to oxygen, for there to be the other elemental diversity so that small rocky planets such as Earth could exist for life to emerge and beings like us to evolve and wonder how we got here, for all of this to be set to happen 13.8 billion years ago, causes this conscious, sentient, and sapient being to wonder if all of this creativity must have come about from purpose. as opposed to purposeless and "undirected processes".
they can believe in their unproveable explanation (the "Multiverse of the Gaps") and i can believe in mine.