What, precisely, is kamma/karma?
Karma is not something one accumulates; this question gets asked so often because, as you say, there is a misconception of it being a 'a solid, substantial' entity.

Karma is volition (intention is a bad translation, IMO), or in abhidhamma, the seven javana citta present in an ordinary mind process (i.e. every moment of experience).

All this means is that every experience you have has the potential to affect future experience in some way; the fact that karma produces a result in kind is indicative of how natural the process is - karma is simply the law of causality that moulds our lives according to our actions. That this should continue after death is not a problem for Buddhism since, as I've noted several places here already, death according to Buddhism is just a concept. The mind states continue to arise and cease according to the law of causality.

As to how one mind can affect the next, I really have no answer besides that through the practice of meditation you can verify for yourself that it is indeed the nature of reality; why it is this way is somewhat irrelevant.