The (orbital) wave function for $l=1$ doesn't just "have to be" antisymmetric. It demonstrably "is" antisymmetric. The relevant part of this wave function is completely determined, it's a particular function, so we may see whether it's symmetric or antisymmetric and indeed, it's the latter.
Two particles – in this case two pions – orbiting each other are described by a wave function of the relative position
$$\vec r = \vec r_1 - \vec r_2 $$
Writing $\vec r$ in spherical coordinates, a well-defined $l,m$ means that the wave function factorizes to
$$ \psi (\vec r) = \psi_r(r)\cdot Y_{lm}(\theta,\phi)$$
The angular dependence simply has to be given by $Y_{lm}$ because $Y_{lm}$ is, up to the overall normalization, the only wave function with the right angular momentum and its $z$-component being $l,m$.
But $Y_{lm}$ is easily seen to be an odd function of $\hat r$ for odd $l$ and even function of $\hat r$ for even $l$. In particular, $Y_{00}$ is a constant while $Y_{10}$ and $Y_{1,\pm 1}$ are proportional to $z$ and $x\pm iy $ on the unit sphere, respectively.
These functions proportional to $x,y,z$ are clearly odd functions of $\hat r$, so they change the sign under $\vec r\to -\vec r$ which is the sign flip equivalent to $\vec r_1\leftrightarrow \vec r_2$. This odd parity is also called antisymmetry.
