Please note that I consider the existence of life to be a coincidence. As such, I do not actually believe that there is a purpose to any form of Life, including Humankind. However, as Humans are capable of assigning a purpose to more or less everything, and I am a Human, I am not ashamed to use the word "purpose" in the title, or anywhere else in the essay, though I don't think I have in the case of the latter anyway.
Humans are unique among the species of Earth. It is true that most traits which we once considered to be ours and ours alone are shared with other species - many apes show a capacity for complicated language, ants could be described as "civilised" (perhaps more so than Humans), dolphins masturbate, etc. But even if we don't have a single unique trait, we have perhaps the most useful mix of rare traits. We have the intelligence and diversity of expressions to form and use complex languages, the structure of our hands gives us the ability to build and manipulate objects in a way most other creatures cannot, and no single other species makes as much use of as many different species as Humans do.
Regardless of how unique we are, we can be certain that, one way or another, Humankind has established itself as the dominant species on Earth. In the grand scheme of things, we have not been the dominant species for long - a few millenia, if that. But we are the first species with the potential to live beyond the existence of Earth itself (though by that time we will have most likely evolved into something else, but for simplicity's sake I will refer to present-day Humans and Post/Trans-Humans as Humans throughout this essay). Due to the caring nature of a large number of Humans, we could also prolong the existence of other species. Indeed, no Apocalyptic Scientific prediction of the fate of the universe took the existence of Humanity, or other sentient lifeforms, into account. We may well be able to preserve the current (desirable) state of the universe indefinitely, and even if we can't it is worth the attempt.
Therefore, I have come to the conclusion that it is Humankind's responsibility to:
1) (Attempt to) Preserve the existence of Life originating from Earth beyond the existence/habitability of Earth itself, preferably eternally.
2) (Attempt to) Preserve the habitability of the Universe, preferably eternally.
Note that other Human-like Lifeforms may exist, whether it be at present or in the future, so even if we fail at the latter, or even the former, it is possible that another species will succeed where we failed.
Also note that I consider non-existence to be better than a joyless or excessively painful existence, so as far as I am concerned Humans should only preserve life on Earth and in the rest of the universe for as long as there is joy, the possibility of joy, no more than a reasonable degree of universally experienced pain and/or the possibility that any excessive and universally experienced pain will be gone.
Finally, we cannot prove that there will be a saviour, so we may as well make ourselves the saviours of the universe, and if there will be a saviour then he/she/it will probably be glad we made the effort.
