A lot of journals are starting to do this. For example PNAS (a top ranking journal) introduced a required "significance statement" in addition to an abstract, which is meant to be understood by a more general audience. So do the PloS journals, and many others.
PLoS says "The goal is to make your findings accessible to a wide audience that includes both scientists and non-scientists", and PNAS says it is meant to "explain the relevance of the work in broad context to a broad readership."