SegWit is a Bitcoin improvement proposal addressing some scaleability and transaction malleability issues. Each Bitcoin transaction consists of inputs outputs and transaction signature. This signature is used by the miners to verify the transaction. Essentially SegWit is a proposal for a software update of the Bitcoin protocol that would allow to separating signatures out of transactions and keeping separated data repository outside the standard block with the signatures. This would making transaction signatures optional in propagation and storage. The SegWit soft fork would solve many problems the Bitcoin network is suffering today.
Segregating the witness allows both existing software and upgraded software that receive transactions to calculate the transaction identifier (txid) of segwit-using transactions without referencing the witness. This solves all known cases of unwanted third-party transaction malleability. Transaction malleability is a problem that makes programming of Bitcoin wallet software more difficult and seriously complicates the design of smart contracts for Bitcoin. (Source: Bitcoin Core)
The transaction signatures alone are accountable for 60% of the block size of the Bitcoin blockchain. Moving signatures out of the transaction block frees up space in the block, thus enabling Bitcoin miners to store more transactions per block. This will make the Bitcoin blockchain faster and cheaper in verifying transactions.
Benefiting Off-chain Protocols
SegWit will pave the way for the development of second layer networks like the Lightning network and other off-chain protocols.
Designing smart contracts for the Bitcoin blockchain is complicated today. Upgrading to SegWit would make micropayment channels and other anticipated new smart contract possibilities less complicated to design, understand, and monitor (Source: Bitcoin Core).
A soft fork is a change to the protocol (set of rules) wherein only previously valid blocks/transactions are made invalid. Since old nodes will recognise the new blocks as valid, a soft fork is backward-compatible. This kind of fork requires only a majority of the miners upgrading to enforce the new rules.