Edge and matter are two completely different Things which just both happen to be happening at about the same time.
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Matter is a new independent third-party standard, which is being adopted by much of the home automation industry, including most of the big players: Amazon, apple, Google, Samsung, Philips, Tuya, Aqara, IKEA, and many more.
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the purpose of matter is to make it easier for end consumers to know whether new products that they want to buy will work with their existing home automation products, in particular, the major voice assistants: Siri, Alexa, Google, and Bixby. And to make installing new products easier.
The idea is you will no longer have to try to figure out on your own whether some new inexpensive Wi-Fi device will work with what you already have. If it has a matter logo, and your platform of choice supports matter, it probably will. so that’s nice. 
Consumer reports has a particularly good matter FAQ:
And there’s a thread in this forum where people are discussing how Matter impacts smartthings
Matter - smart home connectivity standard (formerly Project CHIP)
- smartthings is going to support matter, but not all features of matter. At the time of this writing, it’s basically a one-way integration in. Once matter devices are widely available, you should be able to add pretty much anything with the matter logo to your smartthings account and see it in the smartthings app. No custom code required. But smartthings is not going to support integration outwards: you still won’t be able to add a smartthings hub and its associated devices to, for example, the IKEA app or Apple’s home app.
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There’s a lot more stuff to say about matter, but it will all be happening behind the scenes, so see the matter discussion thread linked to above for those details.
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edge is unique to Smartthings, and it’s just one part of a years long project to get rid of the free groovy hosting that smartthings used to provide. It only applies to devices that talk directly to a smartthings/Aeotec hub. It Is hoped that it will provide better speed and reliability, but we honestly don’t know for sure yet. And it does NOT get rid of the smartthings cloud: smartthings employees have posted in this forum that the system still assumes that the hub is almost always cloud connected. And the smartthings app always requires the cloud. So edge will provide more local processing, but not local-only processing.
So it’s a new smartthings – specific architecture replacing an older smartthings – specific architecture.
Here again is the link to the community FAQ on edge
FAQ: I have no idea what Edge is. Is that a new developer tool? (2022)
- There is a small overlap between edge and matter in that the smartthings method for implementing matter will involve edge drivers for matter devices, just like there are edge drivers for Zigbee devices and edge drivers for Z wave devices. But particularly for Matter all of this should happen behind the scenes: you just add the matter-compatible device to your account and it should automatically find the right edge driver if one is needed.