In general the \mapsto arrow $\mapsto$ is a way of defining a function without naming it. It's similar to the lambda construction in Scheme and Python, and I believe Maple uses ->. So for example, you might define
$$f(x) = x^2$$
or you could write the same definition as
$$f = x \mapsto x^2$$
So the notation
$$F = v \mapsto (x \mapsto f(x,v))$$
from that Wikipedia page could also be written as
$$F(v) = x \mapsto f(x,v)$$
or using the $\cdot$ notation,
$$F(v) = f(\cdot, v)$$
Since $f$ in the Wikipedia article is supposed to be a bilinear form, if you fix one of its parameters (fill in $v$) you get a linear function from vectors to scalars.
Best Answer
It is defined for you in the final equation.
It means the matrix where $0$'s are appended at the bottom.
This is not a standard notation and it is not clear if we always append a single row of $0$'s or could it be multiple rows of zeros.