Short answer: **because it's a complex torus**. Explanation below would take as through many topics.

### Topological covers

The curve should be considered over complex numbers, where it can be seen as a Riemann surface, therefore a two-dimensional oriented closed variety. How to find out whether this particular one is a sphere, torus or something else? Just consider a **two-fold covering** onto $x$-axis and count the Euler characteristics as $-2 \cdot 2 + 4 = 0$ (don't forget the point at infinity.)

### Complex tori

So this is a torus; now a torus with complex structure can be always defined as a quotient $\mathbb C/\Lambda$, where $\Lambda$ is the **lattice of periods**. It can be written as integrals $\int_\gamma \omega$ of any differential form $\omega$ over all elements $\gamma \in \pi_1$. The choice of differential form is unique up to $\lambda \in \mathbb C$.

### Algebraic addition

A complex map of a torus into itself that leaves lattice $\Lambda$ fixed can be only given by a shift. Once you select a base point, these shifts are in one-to-one correspondence with points of $E$. We have unique distinguished point — infinity — so let's choose it as the base point. It follows that we now have an **addition map** $(u, v) \to u\oplus v$, though defined purely algebraically so far.

### Geometric meaning

Now let's stop and ask ourselves: **how to see this addition geometrically**? For a start, consider map that sends $u$ to the third point of intersection with the line containing both $u$ and 0 (the infinity point). It's not hard to see that we fix 0 but change every class $\gamma$ in a fundamental group into $-\gamma$, so we must have the map $u\mapsto -u$ here.

### Group theory laws

What would happen if you took a line through $u$ and $v$? By temporarily changing coordinates so that $u$ becomes the infinity point, one writes down that map as $(u, v) \mapsto -(u+v)$.
Now if you took three points, there would be two different ways to add them; those would lead to $(u+v)+w$ and $u+(v+w)$ as complex numbers, which we **know to be associative**.

### Logically proven

In the above, we worked over complex numbers, but we proved associativity which is a formal theorem about substitution of some rational expressions into others. Since it works over complex fields, it is required to **work over all fields**.

(In any case, the big discovery of mid-20th century was that you actually can take all of the intuition described above and apply it to the case of elliptic curves over arbitrary field)

### Analytic computations (bonus)

Consider a line that passes through points $u$, $0$ and $-u$. This line is actually vertical, and $y$ is a well-defined function there which has two zeroes and one double pole at infinity. After a shift and multiplication of several such functions we'll be getting a meromorphic function on a complex torus with poles $p_i$ and zeroes $z_i$ having the property $\sum p_i = \sum z_i$. This method can give all such functions and only them; it's not hard to see that only meromorphic functions with this property are allowed on elliptic curve.

For example, $\wp'$-functions are the ones that have triple pole at 0 and single zeroes at points $\frac12w_1, \frac12w_2, \frac12(w_1+ w_2)$ where $w_1, w_2$ are generators of $\Lambda$.

### Jacobian of a curve (bonus 2)

The formula above describes what types of functions are allowed on our curve. It is a good idea to organize this information into a curve: in this case, the information is that a single expression $p_1 + p_2 + \cdots + p_n - z_1 - \cdots - z_n$, considered a point of the curve, must vanish. For curves of higher genus, more relations are necessary; for $\mathbb C\mathbb P^1$, no relations beyond *number of poles = number of zeroes* are necessary. Those are relations in the group of classes of divisors (= Jacobian of a curve) mentioned in other answers.

In particular, elliptic curves coincide with their Jacobian and that's another explanation for the additive law.