In short, we do not need to adopt this as an axiom. But...
If there are sets at all, the axiom of subsets tells us that there is an empty set: If $x$ is a set, then $\{y\in x\mid y\ne y\}$ is a set, and is empty, since there are no elements $y$ of $x$ for which $y\ne y$. The axiom of extensionality then tells us that there is only one such empty set.
So, the issue is whether we can prove that there are any sets. The axiom of infinity tells us that there is a set (which is infinite, or inductive, or whatever formalization you use). But this seems like a terrible overkill to check that there are sets, to postulate that there are infinitely many.
Some people prefer to have an axiom that states that there are sets. Of course, some people then just prefer to have an axiom that states that there is an empty set, so we at once have that there are sets, and avoid having to apply comprehension to check that the empty set exists.
Other people adopt a formalization of first order logic in which we can prove that there are sets. More carefully, most formalizations of logic (certainly the one I prefer) prove as a theorem that the universe of discourse is nonempty. In the context of set theory, this means "there are sets". This is pure logic, before we get to the axioms of set theory. Under this approach, we do not need the axiom that states that there are sets, and the existence of the empty set can be established as explained above.
(The logic proof that there are sets is not particularly illuminating or philosophically significant. Usually, one of the axioms of first order logic is that $\forall x\,(x=x)$. If $\exists x\,(x=x)$ --the formal statement corresponding to "there are sets"-- is false, then $\forall x\,(x\ne x)$. Instantiating, we obtain $x\ne x$, and instantiating the axiom $\forall x\,(x=x)$ we obtain $x=x$, and one of these conclusions is the negation of the other, which is a contradiction. This is not particularly illuminating, because of course we choose our logical axioms and rules of instantiation so that this silly argument can go through, it is not a deep result, and probably we do not gain much insight from it.)
It turns out that yet some others prefer to allow the possibility that there are empty universes of discourse, so their formalization of first order logic is slightly different, and in this case, we have to adopt some axiom to conclude that there is at least one set.
At the end of the day, this is considered a minor matter, more an issue of personal taste than a mathematical question.
Best Answer
This is a historical issue really.
Originally set theory was developed by Cantor and the well ordering principle was somewhat assumed in the background (e.g. Cantor's proof of the Cantor-Bernstein theorem was a corollary from the fact that every two cardinalities are comparable).
In 1904 Zermelo formulated the axiom of choice and proved its equivalence to the well ordering principle. He later formulated more axioms which described our intuition about sets, therefore removing the "naivity" from the Cantorian set theory. He did not add the axiom of foundations, nor the schema of replacement. Those were the result of Skolem and Fraenkel which were popularized by von Neumann.
The axiom of choice remained controversial, the thought that the continuum can be well-ordered was mind boggling and caused many people feel uneasy about this axiom. Further results like the Banach-Tarski paradox did not help to accept this axiom either.
Prior to set theory most mathematics was somewhat constructive in the sense that things were finitely generated or approximated by finitary means (e.g. limits of sequences). It requires quite the leap of faith to go from things you can pretty much write down to things which you cannot describe but only prove their existence. In this sense the axiom of choice augments the way we do mathematics by allowing us to discuss objects which we cannot describe in full.
It was questionable, therefore, whether this axiom is even consistent with the rest of the axioms of set theory. Gödel proved this consistency in the late 1940's while Cohen proved the consistency of its negation in the 1960's (it is important to remark that if we allow non-set elements to exist then Fraenkel already proved these things in the 1930's).
Nowadays it is considered normal to assume the axiom of choice, but there are natural situations in which one would like to remove it or find himself in universes where the axiom of choice does not hold. This makes questions like "How much choice is needed here?" important for these contexts.
Some things to read: