The ‘Oligo-Miocene’ part
The Oligocene and the Miocene are epochs of geological time. The Oligocene lasted from ~33.9 million years ago to ~23 million years ago; the Miocene followed immediately after the Oligocene, and lasted until ~5 million years ago.
Oligo-Miocene refers to events that happened around the boundary between the two epochs at ~23 million years, or that happened throughout the Oligocene and Miocene epochs.
The ‘radiation’ part
‘Radiation’ here is evolutionary radiation, a biological term usually referring to an increase in the number of species or in the differences between them. The Palaeontology Online glossary defines ‘evolutionary radiation’ as follows.
An evolutionary radiation is an increase in either taxonomic diversity (i.e. the number of species) or morphological disparity (how they differ in anatomy). This can result from an adaptive change or the opening of ecospace.
Putting the two parts back into the contexts where you found them: the first article mentions an increase in the diversity or marine mammals, and the second in the diversity of dry-adapted shrubs. Both these increases occurred between the start of the Oligocene and the end of the Miocene.