To avoid ambiguity, this question pertains to the construction of Lagrangian densities (including interaction terms) in terms of their values at single points in spacetime.
In classical mechanics in which interactions take place between point-like objects I can see that the motivation for demanding locality is that objects should only be able to directly interact (i.e. without any intermediary medium to transmit the interaction) if they are in direct contact (i.e physically "touching" - they at the same point or infinitesimally close to one another). In this case locality is a philosophically (and intuitively) desirable property, but not essential, as there is no upper limit on which information can propagate. Similarly, locality in time is a desirable property as well, as it would be impossible to predict evolution of physical systems if their dynamical state at a given instant depended on their dynamical state at other instants in time (other than those infinitesimally close to the interval that one is considering).
However, when we consider relativity, locality of interactions becomes essential, as there is a finite upper bound on which information can propagate and therefore instantaneous interactions between spatially separated objects is simply not possible.
Generally though, locality in space and time means that one can describe physical interactions as a result of objects situated at a point, or in a sufficiently small neighbourhood of that point.
This being the case, why is it that in field theory we require interactions to occur at single spacetime points? Is it simply because if we didn't then in different frames of reference the interactions between fields could occur at more than one spacetime point (due to the mixing of spatial and temporal coordinates in Lorentz transformations between different frames), thus violating Lorentz invariance and also meaning that the interaction will will not be local in all inertial frames of reference?