Bilinearity: what does it mean?

It means linear on the left:

$$\langle au + bv, w \rangle = a \langle u, w \rangle + b \langle v, w \rangle$$

and on the right:

$$\langle u, av + bw \rangle = a \langle u, v \rangle + b \langle u, w \rangle.$$

By symmetry, linearity on the left implies linearity on the right, so that's why right-linearity isn't explicitly mentioned (although strictly speaking it should be).

By currying, you can think of a function of two variables, $f\colon X\times Y \to Z$, as a composition of two functions: first, given $x\in X$, you have a function that sends $x$ to the function $f_x\colon Y\to Z$; and then this function is evaluated at a $y\in Y$ to give $f_x(y) = f(x,y)$.

When $X$, $Y$, and $Z$ are vector spaces, the set of functions from $Y$ to $Z$ (written $Z^Y$) is also a vector space, so the notion of linearity makes sense for the set of all such functions.

The function $f$ is "bilinear" if and only if both the map $X\to Z^Y$ and the maps $Y\to Z$ that we get are linear. That is, if and only if $f_{x+\alpha x'} = f_x+\alpha f_{x'}$ for all $x,x'\in X$, and scalar $\alpha$; and for each $x\in X$ the map $f_x\colon Y\to Z$ is linear. So $f$ itself is obtained by working with two linear functions.

(Symmetrically, you can think of $f$ as given first by a function that takes $y\in Y$ to a function $g_y\colon X\to Z$, and then this function is evaluated at $x\in X$ via $g_y(x) = f(x,y)$. Again, $f$ is linear if and only if the function $g\colon Y\to Z^X$ is linear, and the functions $g_y\colon X\to Z$ are each linear.)

Alternatively, notice that $f$ is a function of two variables. Bilinearity is precisely the condition "linear in each of the variables separately". So you have a function which is linear in two distinct ways: in the first variable, and in the second variable.

(Just wait until you get to the notion of sesquilinear funtions; trying to interpret "one-and-a-half linear" will give you a headache).

Also remember that not all nomenclature is necessarily accurate-as-to-intuition, and that the meaning of some words has sometimes changed from the time in which the nomenclature because fixed and present day.