SegWit transactions enhance the security of the Bitcoin network by enforcing signature checks before any other checks. This prevents certain types of attacks and makes the network more resistant to vulnerabilities that exploit transaction malleability. Besides, SegWit's introduction was a soft fork, meaning that it did not require a majority consensus of miners to activate. This made the upgrade process smoother and allowed for a more gradual transition.
And here is a link for beginner to learn Segwit, hope it will be helpful.
https://blockgeeks.com/guides/what-is-segwit/The transaction malleability wasn't due to the failure of checking for signatures, but it segregated the signature from the actual transaction itself. This prevents the signature from being modified and thus being malleable.
It was anything but smooth. It was totally unnecessary for it to be dragged on for longer than it did. Segwit's activation requires miners to signal support for it as well and it definitely required a super majority of the miners to signal support for it to be activated. In fact, most if not all soft forks work in this manner. USAF was not materialized and it was definitely not activated by user signalling. Any forks should have consensus and majority support.