I was doing some research about the ways to cut out transaction fees when I came across SegWit. I discovered that SegWit was introduced to fix transaction malleability and improve block space efficiency, but I’m still a bit unclear and confused on how it works under the hood.

Could someone help and explain what exactly changes in a SegWit transaction compared to a legacy one? How does this result in lower transaction fees? Why does it fix the transaction malleability issue?

And also, some examples comparing a legacy vs. SegWit transaction format would be really helpful.