I'm not a coder but I'll take a crack at it..

Typically from what I see a legacy addy starts with a 1 and a segwit addy starts with a 3.. Not that this is the end all be all..
As far as I understand, segwit vs legacy transactions use a different code to wright transactions so the data that must be saved in the blockchain to confirm these segwit transactions is more compressed or just smaller than the amount of data you have to save in the blockchain for a legacy transaction to confirm the same/similar transaction.. 

You save on miner fees paid in this way because fees are based on how much data you have to write to the blockchain, or space taken up in a block..
Space is very limited so the less space you take up with your TX the less you have to pay and the more TXs will fit per block in total..
This makes fees paid per TX cheaper and more room for more TXs per block therefore higher throughput capacity total for all of Bitcoin.. Though we haven't scene a very big change because the space savings aren't drastic and hasn't yet been thoroughly adopted by users (including myself)..

I haven't really been sending around a lot of BTC lately due to craziness and the places I use are all still using legacy addys so to be honest I have not used segwit yet..
When I have the option to choose my fees paid I use a somewhat low fee as compared to the current going rate and supplement them with the accelerators available..



You are right but it is quite technical, you can only get a layman's explanation from me