Since this question, Arkani-Hamed among others has published the text, Grassmanian Geometry of Scattering Amplitudes which elucidates in a pedagogic fashion the amplituhedron approach.
To fully understand the derivation, there are many preliminaries to be understood, but I will try to present a relatively self-contained computation of an amplitude, but this requires accepting some results a priori. The book Scattering Amplitudes by Elvang et al is also a good reference.
The key figure in the amplituhedron approach is the Grassmannian $\mathrm{Gr}(k,n)$ which is the space of all $k$-planes in an $n$-dimensional space. One has as elements the $n\times k$ matrices $C_{al}$ which are $k$ $n$-component vectors that define the plane in $\mathbb C^n$.
However, note in the spinor helicity formalism, by specifying momenta as $p_{\alpha \dot \alpha} = \lambda_\alpha \bar{\lambda}_{\dot \alpha}$, we can make the rescaling $\lambda_\alpha \to t_\alpha \lambda_\alpha$ and describe the same momenta, so we have a $GL(1)$ redundancy. For the Grassmannian, this means we consider any non-degenerate linear transformation to describe the same plane, and so we have a $GL(k)$ redundancy.
The key to evaluating amplitudes is the unique cyclically invariant integral that generates all Yangian invariants, given by,
$$\mathcal L_{n,k}([i|, |i\rangle, \eta_i) = \int\frac{d^{n \times k} C}{GL(k) \prod_{j=1}^n M_j} \left[ \prod_{a=1}^k \delta^2 \left(\sum_i C_{ai}[i| \right) \delta^{(4)} \left(\sum_{i}^n C_{ai}{\eta_i}_A\right)\right] \times$$
$$\times \left[ \prod_{a'=k+1}^n \delta^2 \left(\sum_i \tilde C_{a' i} \langle i|\right)\right]$$
where $M_j$ are the minors, and we specify the external data as $([i|, |i\rangle, \eta_i)$ in momentum space where $\eta_i$ are the superpartners and $\tilde C$ is the complement of $C$. It turns out this $\mathcal L_{n,k}$ is the cyclic invariant integral over all $k$-planes in the Grassmannian.
The delta function constraints geometrically signify that the $2$-plane spanned by $[i|$ is orthogonal to the $k$-plane $C$ and similarly the plane defined by $|i\rangle$ is orthogonal to the $(n-k)$-plane $\tilde C$. Since $C,\tilde C$ are complements, they each respectively contain the planes spanned by $[i|$ and $|i\rangle$. From the geometry, we end up getting momentum conservation for free:
$$\sum_{i=1}^n |i\rangle [i| = 0.$$
Let's now compute $\mathcal L_{n,2}$. For $k=2$ we identify the planes specified by the matrices $C$ as those spanned by the momenta data. We have,
$$\begin{pmatrix}
C_{11} &C_{12} & \dots & C_{1n}\\
C_{21} & C_{22} & \dots & C_{2n}
\end{pmatrix}
=
\begin{pmatrix}
|1\rangle^i &|2\rangle^i & \dots & |n\rangle^i\\
|1\rangle^{\dot 2} & |2\rangle^{\dot 2} & \dots & |n\rangle^{\dot 2}
\end{pmatrix}.$$
The bosonic delta function encodes $\delta^{(4)}(P) = 0$ and the Grassmann delta function encodes conservation of the supermomenta, $\delta^{(8)}(\tilde Q) = 0$. The minors are,
$$M_i = \epsilon^{ab}C_{ai}C_{b,i+1} = -\epsilon_{\dot a \dot b} |i\rangle^{\dot a} |i+1\rangle^{\dot b} = -\langle i, i+1 \rangle$$
and combining all the factors yields,
$$\mathcal L_{n,2} = (-1)^n \frac{\delta^{(8)}(\tilde Q) \delta^{(4)}(P)}{\prod_{i=1}^n \langle i, i+1 \rangle}$$
which is the tree-level MHV amplitude, $\mathcal A_{n}^{\mathrm{MHV}}$, written in the compact spinor-helicity formalism.
