From what I understand Feynman reasons that since we can have negative numbers (-5 apples), why not have assign negative values of probabilities? He takes the total probability formula (Eq. 1) and assigns negative values to the conditional probabilities (small p's). Conditioning simply means that one is counting the events of interest only in the sub-set of all possible events, which meets this condition - let's call it A after Feynman. However, if this conditional probability is not positive, there are no such events in the subset (they do not occur in situation A). Since we know that they can occur anyway (because Feynman is talking about them), they must be somewhere outside of this subset - in some another subset, i.e. under another condition, call it B. Those subsets add up to the full set of possible events and they don't overlap - otherwise Feynman couldn't have used the total probability equation. So where is Nemo? (Nemo is dead - overfishing.) In which condition should I add those events in the total probability formula - the one under which they can't occur or the one under which they can occur? And if I also have conditions C, D, ..., should I treat events as positive probabilities in respective subsets or rather as negative probabilities in the complementary set? Equivalently, which subsets are the pre-images of the Borel sets on which I defined the probability in the real space? I wonder how Feynman magic clarified that, befores he moved on to using ...
Another explanation of Feynman's idea is even more daunting: he seems to assume that if probability is a type of a measure and there exist signed measures which can be have negative values, then probability can be negative too. Measure can indeed be extended to negative numbers, to complex numbers, vectors, Borel alpacas or whatever one wishes as long as it's additive and consensual. Probability is indeed a type of a measure. Howeverm, not all measures are probabilities. This paper is so wrong that it's dificil to argue.
On a side note, there are situations in which negative probabilities crop up in models of natural phenomena - it's only a signal that there is more to them than a simple theoretical model or our understanding assume. Most physical systems obey the assumptions of the Kolmogorovian probability theory (Kolomogorov isn't saint either - he cut corners of mathematical rigour when introducing his calculus, but it's a different story). However, sometimes we cannot fulfil these assumptions (e.g. the statistical stabilisation or reproducibility assumptions) and it leads to anomalies - death of reality, Bell inequalities, ghost particles, etc. But they witness barely the limit of our mathematical map of nature rather than represent real physical problems.
Negative energies (somehow mentioned on this occassion) of bound states simply mean that you need to perform some work/provide energy to the system to change its state to a one with positive energy. For example an electron in a hydrogen atom has much lower energy compared to a free electron with respect to its centre of mass with a free proton. If you want the latter as zero on the energy scale and then starts to bring the two particles closer to each other, their energy will go down the negative scales, because they will be bound by the Coulomb interaction, which attracts them - and in the darkness binds them (-: Who's having LOTR maraton for X-mas? I'm watching The Witcher this year and I think it's great.