What does $ds^2$ mean and how does it specify a metric?

Let $H$ be the upper half-plane in $\mathbb{R}^2$. How does the following expression $$ds^2= \frac{dx^2+dy^2}{y^2}$$ specify a Riemannian metric on $H$?

I don't understand what the expression means. If the $y^2$ wasn't there, then $g=dx^2+dy^2=dx\otimes dx+dy\otimes dy$ is the standard metric. This does make sense to me: at every point it associates the standard inner product. But the above expression is not even symmetric in $x,y$ so it isn't meant to be read the same way, but I don't know how to make sense of it.

As you note, $dx^2 + dy^2$ means $dx \otimes dx + dy \otimes dy$. The expression means exactly what it says: you divide this tensor by $y^2$. It may help to write it as

$$ \frac{1}{y^2} dx \otimes dx + \frac{1}{y^2} dy \otimes dy $$

While a metric needs to be symmetric, there is no reason it needs to be symmetric under swapping $x$ and $y$. Let me add color to help you through your confusion: the tensor we define is

$$ g = \frac{1}{y^2} \color{blue}{dx} \otimes \color{red}{dx} + \frac{1}{y^2} \color{blue}{dy} \otimes \color{red}{dy} $$

The requirement that $g$ be symmetric means that $g$ needs to be equal to

$$ \frac{1}{y^2} \color{red}{dx} \otimes \color{blue}{dx} + \frac{1}{y^2} \color{red}{dy} \otimes \color{blue}{dy} $$

and it is, because the color is just notation and isn't part of the metric.

An example of a tensor that isn't symmetric is $dx \otimes dy$.

A "fundamental form" $$ds^2=E(x,y)\>dx^2+2F(x,y)\>dx\>dy+G(x,y)\>dy^2\ ,$$ in your case $$ds^2={dx^2+dy^2\over y^2}\ ,$$ tells you how to measure the length of a curve $\gamma$ given in the form $$\gamma:\quad t\mapsto\bigl(x(t),y(t)\bigr)\qquad(a\leq t\leq b)\ .$$ The rule is: Write formally $$dx=\dot x(t)\ dt,\quad dy=\dot y(t)\ dt$$ and compute $$L(\gamma)=\int_a^b\sqrt{E\bigl(x(t),y(t)\bigr)\dot x^2(t)+2F\bigl(x(t),y(t)\bigr)\dot x(t)\dot y(t)+G\bigl(x(t),y(t)\bigr)\dot y^2(t)}\ dt\ ,$$ in your case $$L(\gamma)=\int_a^b{\sqrt{\dot x^2(t)+\dot y^2(t)}\over y(t)}\ dt\ .$$ The intuitive idea is the following: For reasons particular to the problem at hand the length of an "infinitesimal segment" connecting the points $(x,y)$ and $(x+dx,y+dy)$ should not be the usual pythagorean $\sqrt{dx^2+dy^2}$; instead it should depend in a still "natural", but more complicated way on the vector $(dx,dy)$. In your case the length of this "infinitesimal segment" should be its euclidean length, divided by the $y$-coordinate of the point where it is located.

It can help to fall back on an extrinsic point of view to get some intuition.

Let $S$ be a Riemannian manifold of dimension 2, smoothly embedded in $\mathbb R^n$. We can associate with each point in $S$ a vector in $\mathbb R^n$. Of course, $\mathbb R^n$ has an inner product, but we naturally do not expect the inner product of two points in $S$ to tell us anything about the geometry of $S$.

When you introduce a pair of coordinates to parameterize $S$, you can talk about the tangent space. Let $s$ be the position of some point in $S$, then the vectors in the tangent space taken on the form

$$e_x = \frac{\partial s}{\partial x}, \quad e_y = \frac{\partial s}{\partial y}$$

Again, $s$ is a vector in $\mathbb R^n$, so these expressions have a sensible, concrete meaning.

The metric, then, merely tells us about the vectors in the tangent space and how they are related to each other. You can write it as

$$ds^2 = (e_x \cdot e_x) dx^2 + (e_y \cdot e_y) dy^2 + 2 (e_x \cdot e_y) dx \, dy$$

So the metric tells us whether the tangent space vectors are orthogonal to one another, and about their lengths. This is the basic extrinsic viewpoint; intrinsically, one has to work a little harder, but hopefully this is enough to gain some intuition.