For a start, I guess that one should be very familiar of the proofs of Furstenberg's diophantine result, as this paper generalizes this theorem.
Secondly, it might be of interest that Zhiren Wang in his PhD thesis generalized Berend's results to an effective proof (in a method which is some-what similar to the method which Bourgain-Lindenstrauss-Michel-Venkatesh used to generalize Hillel results).
$(3)$ is easy, just like in Furstenberg's theorem, otherwise, the whole semi-group would be contained in a semigroup spanned by one element, and it is well-known that such semigroups (in the case of diagonal actions, in contrary to Ratner's theorem) do not enjoy measure rigidity.
This is somewhat equivalent to the assumption of higher-rank in the Einsiedler-Katok-Lindenstruass theorem about rigidity of higher-rank diagonalizable actions.
$(2)$ roughly corresponds to the fact that the system is expansive system.
Otherwise (as shown by Berend in his paper about necessity of the conditions) - you can find some small set "which won't grow", due to the fact that every matrix will contract the set and won't spread it out.
$(1)$ is more mysterious, and as Dani sates in his paper, it implies that the action is basically a diagonal action (actually conjugated to a diagonal one, but it won't effect the rigidity).
The other question should be answered in another topic I guess, but Ian said the condition more or less, if you are familiar with Sinai/Pesin's formula for entropy (or in this case, the geometry of Markov partitions from the Adler-Weiss theorem), you can massage the proof a bit to get a proper condition (assuming some conditions on the matrices, say commutation relations). I believe that the formal proofs appear in Walters or in the new Einsedler-Ward volume about entropy.
