I think the original motivation was to remove signatures from the data that is hashed so as to make the hash of the transaction (the TX ID) orthogonal to the signature data, so as to deal with malleability since due to the use of ECDSA there are two versions of the same signature that are equivalent (one of the reasons Wuille says he wants to replace them Schnorr signatures instead).
But then to do what they are calling a "segregated witness", the security model changes from every node verifying every detail for themselves, to every node assuming that some node will publish a proof-of-cheating if any activity was incorrect.
They are suggesting nothing of the sort. Full nodes in Bitcoin will still download the entire chain, including signatures. The peer-to-peer protocol will expect the signatures to be delivered along with the block and will then verify it using a hash stuffed in the coinbase.
They are suggesting to
add a new sort of less-than-full node that is less secure than full nodes, but full nodes will operate under the same security model, just using a different method for fetching (and verifying) the signatures.
Apparently BBR is including the signatures in the hash of the TX ID.
It does not. BBR neither includes the signatures in the TX ID nor does it include an additional hash.
This is the interesting part, in that gmaxwell claims this introduces some sort of vulnerabilities, but it isn't clear to me what they are.