A lot of the work on smooth manifolds is to let us use Euclidean analysis to merely locally Euclidean things that come up. Things that aren't Hausdorff are terrible and scary, real intuition busters (at least in my case), so I don't mind at all that we require that. And in fact, with just these 3 requirements (and a smoothness requirement), smooth manifolds can actually be viewed more or less in real space (the Whitney Embedding Theorem). And our real space intuition is pretty good, you know? And it suffices, has really nice properties, provides some pretty rich material, etc.
But there are some people who care a lot about non-Hausdorff manifolds. In fact, secretly, we see them and don't know it. The etale space of the sheaf of continuous real functions over a regular manifold is a manifold, and it's sometimes non-Hausdorff.
Similarly, there are people who care about non-second-countable manifolds. But these are also a bit unfortunate. One of the great things about second countability is that it guarantees that ordinary manifolds are paracompact. A paracompact smooth manifold admits partitions of unity subordinate to a refinement of any cover. Why is this important? (for that matter, what does it really mean?) While it's easy to stitch together continuous functions to make a continuous function (just sort of join the ends together, right?), it's really hard to stitch smooth functions together in general (join the ends together, and perturb it so the first derivatives align, and so the second align, etc.). But this can be done with little fuss with partitions of unity, and thus with little fuss with second-countability.
And if you study smooth manifolds, you'll see that partitions of unity are immediately used for everything.
So it's the way it is because it has these really nice properties, right? Well, why don't we just require manifolds to be paracompact? (Firstly, there is a distinction, but it's 'small.' A manifold with more than countably many disconnected components may be metrizable, and thus paracompact, but obviously won't be second-countable). In this case, the category of paracompact manifolds is closed coproducts, which doesn't hold for second-countable manifolds. In fact, some people do only consider paracompact manifolds.
At the end of the day, Hausdorff and second countability are exactly what let us use the embedding theorem to view manifolds in real space, and that's what's deemed important for people on their first tour through manifolds.
As written, the term "locally Euclidean" is in fact not even defined at all (only "locally Euclidean of dimension $n$" is defined). What it appears the author really intended is the following pair of definitions:
Definition 5.1'. A topological space $M$ is locally Euclidean of dimension $n$ at a point $p\in M$ if $p$ has a neighborhood $U$ such that there is a homeomorphism $\phi$ from $U$ onto an open subset of $\mathbb R^n$. A topological space $M$ is locally Euclidean if for each $p\in M$, there exists $n$ such that $M$ is locally Euclidean of dimension $n$ at $p$.
Definition 5.2'. A topological manifold is a Hausdorff, second countable, locally Euclidean space. It is said to be of dimension $n$ if it is locally Euclidean of dimension $n$ at every point.
I would add, however, that this definition is not very standard. Most people define manifolds such that they must have the same dimension at every point, even if they are disconnected.
Best Answer
There are non-Hausdorff spaces that are locally Euclidean; some people include them in the class of manifolds, and some prefer to exclude them by requiring a manifold to be Hausdorff. The line with two origins is a simple example of a non-Hausdorff manifold: it is locally Euclidean, since it has a base of open sets homeomorphic to $(0,1)$, but the two origins cannot be separated by disjoint open sets.