The intuition of the positivity condition can be explained through the following analogy with Morse theory. Given a compact oriented manifold $M$ with a Morse-Smale function $H$, then

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  The degree zero (resp. $n$) elements in the Morse cochain complex are generated by the local maxima (resp. local minima) of $H$ whose unstable submanifolds (resp. stable submanifolds) have preferred orientations.

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  The fundamental class $1\in H^{0}(M)$ is represented by the sum of the local maxima with the preferred orientation. Similarly with the generator of $H^{n}(M)$.

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  A generic point on $M$ lies on exactly one Morse flowlines which starts with one local maximum point and ends on one local minimum point. More formally, if we construct the universal family of Morse flowlines that start with some local maximum and ends on some local minimum, the evaluation map would sweep out the fundamental cycle of $M$ as an $n$-dimensional current, without any cancellation effect.

Now for simplicity, if we start with an embedded Lagrangian brane $L$, and preform a small generic Hamiltonian deformation $\phi_{H}(L)$, then the Floer cohomology $HF^{*}(L,\phi_{H}(L))$ is computed as the Morse cohomology, so the Morse theory statements above would imply the positivity condition at least in such special cases. The intuition is that if the Lagrangian $L^{\prime}$ (together with its brane structure) is a sufficiently small deformation of $L$, then we expect the positivity condition to hold for the bordism current between $L$ and $L^{\prime}$.