For 2D manifolds the scalar curvature is the Gauss curvature. Positive means the manifold closes in on itself, negative means it spreads out like a saddle.
Some of this parallel remains in higher dimensions, since the sign of curvature determines the asymptotic volume comparison for geodesic balls (positive curvature $\implies$ less-than -Euclidean volume, negative $\implies$ greater than Euclidean). However this is on small scale only, and therein lies a problem: the scalar curvature does not deliver much global geometric information.
Indeed, the asymptotic volume definition implies that scalar curvature is additive under products. Thus, by taking product with a sufficiently scaled-down compact surface of constant negative curvature (think of a tiny double torus), we can make scalar curvature negative, as long as it was bounded to begin with. And by taking product with a tiny sphere, the scalar curvature can be made positive.
These examples explain why there isn't a picture of what a manifold of positive/negative scalar curvature looks like: it can look like pretty much anything. (There are some obstructions to manifolds admitting positive scalar curvature, but they are not easy to visualize.)