Basically, all scriptPks (output scripts) become something like [hash] OP_TRUE inside the block and when another tx references them as an input the true scriptSig is stored outside the block, and a hash of all the block's signature data is stored somewhere within the block. Blocks are transmitted with their extra data, and inside the block all the signatures are blank.
edit: If I've helped you understand a bit better, please upvote my post here too (they're np, if you want to upvote you need to remove that).
https://np.reddit.com/r/Bitcoin/comment ... of/cxptrr9
The original implementation of segregated witness was also CryptoNote coins, as I mention here. Wuille's presentation fails to credit this alternative cryptocurrency at all (as did the Elements sidechains presentations/implementation/notes). Ironically the second time a major Bitcoin soft fork has come from an alt.
https://np.reddit.com/r/Bitcoin/comment ... of/cxpxi5t
It makes fraud proofs easier, which is ultimately one of the most promising scaling strategies: Not everybody checks the whole blockchain, but assuming someone is checking every part of it, and you are able to receive communications from that someone, you're almost as secure as if you were.
It reduces the amount of data you need to download in some circumstances
To the extent that it's adopted, it moves the content of blocks out of the jurisdiction of the 1MB limit, which is equivalent to raising the limit, but by doing it non-explicitly it avoids setting a precedent, and by using a soft fork it works around the extreme consensus requirements some people have tried to set up for hard forks.
It's an amazing political solution with something in it for all the most opposed people: Malleability fixes are need for Lightening, separating signatures allows /u/nullc to oppose future block size increases without dynamiting his own (very cool) Confidential Transactions work which has bigger signatures, and it should be easy to persuade /u/luke-jr not to make a big voting war like he did over p2sh since the soft fork technique was his idea.
Sources:Basically the same as BIP16/P2SH, except you put the signature data outside of the transaction and merge-mine those commitments.
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