In classical electrodynamics, properties of point charges have to be inferred from extended charge distributions, in the limit of zero size.
Moreover, classical electrodynamics is fully compatible with special relativity. That being the case, the "real" explanation should not involve action at a distance. Instead, each part of an extended charge distribution interacts with the field in its immediate vicinity, disturbances in the field propagate through space, and in this way, charges are able to act on other charges.
If the amount of momentum taken up by the field when it interacts with one part of an extended charge distribution is not equal and opposite to the amount of momentum taken up by the field when it interacts with some other part of the extended charge distribution (or the same part at a later time) then there will be a net transfer of momentum from the charge distribution to the field—the radiation reaction.
Since the field is the means by which charges exert forces on other charges, the aforementioned description of the phenomenon is not physically different from the one in which the forces exerted by different parts of the charge distribution on each other are calculated directly without mentioning the fields.