The Segregated Witness soft-fork (segwit) includes a wide range of features, many of which are highly technical. This page summarises some of the benefits of those features.
Malleability Fixes
Bitcoin transactions are identified by a 64-digit hexadecimal hash called a transaction identifier (txid) which is based on both the coins being spent and on who will be able to spend the results of the transaction.
Unfortunately, the way the txid is calculated allows anyone to make small modifications to the transaction that will not change its meaning, but will change the txid. This is called third-party malleability. BIP 62 (“dealing with malleability”) attempted to address these issues in a piecemeal manner, but was too complicated to implement as consensus checks and has been withdrawn.
For example, you could submit a transaction with txid ef74…c309 to the network, but instead find that a third-party, such as a node on the network relaying your transaction, or the miner who includes your transaction in a block, modifies the transaction slightly, resulting in your transaction still spending the same coins and paying the same addresses, but being confirmed under the completely different txid 683f…8bfa instead.
More generally, if one or more of the signers of the transaction revise their signatures then the transaction remains valid and pays the same amounts to the same addresses, but the txid changes completely because it incorporates the signatures. The general case of changes to signature data (but not the outputs or choice of inputs) modifying the transaction is called scriptSig malleability.
Segwit prevents third-party and scriptSig malleability by allowing Bitcoin users to move the malleable parts of the transaction into the transaction witness, and segregating that witness so that changes to the witness does not affect calculation of the txid.
Who benefits?
Wallet authors tracking spent bitcoins: it’s easiest to monitor the status of your own outgoing transactions by simply looking them up by txid. But in a system with third-party malleability, wallets must implement extra code to be able to deal with changed txids.
Anyone spending unconfirmed transactions: if Alice pays Bob in transaction 1, Bob uses that payment to pay Charlie in transaction 2, and then Alice’s payment gets malleated and confirmed with a different txid, then transaction 2 is now invalid and Charlie has not been paid. If Bob is trustworthy, he will reissue the payment to Charlie; but if he isn’t, he can simply keep those bitcoins for himself.
The Lightning Network: with third-party and scriptSig malleability fixed, the Lightning Network is less complicated to implement and significantly more efficient in its use of space on the blockchain. With scriptSig malleability removed, it also becomes possible to run lightweight Lightning clients that outsource monitoring the blockchain, instead of each Lightning client needing to also be a full Bitcoin node.
Anyone using the block chain: smart contracts today, such as micropayment channels, and anticipated new smart contracts, become less complicated to design, understand, and monitor.
Note: segwit transactions only avoid malleability if all their inputs are segwit spends (either directly, or via a backwards compatible segwit P2SH address).