ELI5 (ok, maybe 10) Matrix chat

No, you are right, Matrix is a protocol. Some of its major points are:

• It's decentralized (There are homeservers, but federation is built into the protocol. Think of this like email servers. There are many different servers out there, and they can talk to each other. Or not, depending of the configurations the server admins set.)
• It's a interoperable protocol, allowing users to communicate across different service providers and platforms. This is why Beeper used it as it's core.
• • Bridges allows taking posts from one service and posting it to another, and vice versa. For example, users from Discord, IRC, Telegram, etc can all talk between each other with little effort from whatever platform they choose. They don't need to be using a matrix server themselves, just someone who has set up a bridge with one in their spaces.
• • Puppeting allows users using a matrix homeserver & a bridge that supports puppeting to log in with their other service's account. For example, if you were to do this with Discord , you'd then be able to have all over your messages, including dms, bridge to your "matrix" account. You'd also be able respond via your account directly. That is, if you send a message to Discord from your matrix homeserver, it'll get posted via your Discord account and not the Discord bot used for the normal bridging stuff. NOTE: Puppeting is against Dscord's TOS, but is use in this example as most know what it is.
• End-to-end encryption is built heavily into the protocol. Not every space has to have it, but it's a core feature and enabled by default for DM's.
• It's an open source protocol

Personally, I feel there's still a few cons:

• Is is not polished.
• The new user onboarding needs work
• Some QOL things are missing. Thought this is more an issue with Element, the default reference client. They have a very shrill default notification sound and no way to easily change it. There's an open issue with a fair few users who understandable want to be able to change it easily.
• The spec is evolving still, and there's some things that haven't been figured out yet.

In saying that, I do still use Matrix and run a server myself. I think it's good, but needs some work still.

As to a NZ homeserver or space, I'm currently unaware of any sorry. And GoT felt like it was going of the rails before the last season >.<

Matrix does have some good documentation though, you'll find a lot of info on their site here: https://matrix.org

Matrix is an open source protocol for federated chatrooms, kind of like if someone mixed Lemmy with Discord and Signal. You make an account with a Matrix 'homeserver' (can be your own self-hosted server) just like Lemmy, so you'll have a username like user@homeserver.com. Once you do you can join any number of Spaces, akin to Discord servers. Unlike Discord, these servers will be hosted on the homeserver which means they can be self-hosted, and often come with strong safety guarantees like end-to-end encryption or the double ratchet algorithm as seen in Signal (depending on how the homeserver is configured). Matrix is really just a protocol, so there are a bunch of chat clients that implement it, the first-party client is called Element, but there are many to choose from.

I would argue the main reason to use Matrix over Slack or Discord is much the same reason you'd use Signal over Whatsapp - data privacy. Because you can self-host the homeserver any spaces you make can be hosted on your local machine. For those who are privacy advocates that's a very good reason to use Matrix over most other solutions. If you're a company or a concerned individual that routinely deals with data that really shouldn't be on the 'cloud' (e.g. trade secrets, materials under NDA, personal information, etc.) then Matrix seems like a better fit than say Slack, provided you self-host. Discord has been under fire for their privacy policy for end users, so you might consider Matrix as a replacement for Discord too.

On a related note, this is the first I'm hearing of beeper.

Is the main draw just being able to combine messaging across different apps into one, or are there other benefits too?

I'll try to add on to what others have said. Matrix is a protocol. Element is the client made by the people behind Matrix but you can use any Matrix client. Matrix is the protocol behind Beeper but they don't put much effort into integrating with Matrix, so for example (last I checked) you can't log in to a Matrix account on Beeper.

However, your Beeper account is a Matrix account so you can join Matrix rooms with it. It's a little janky, but basically you go to the menu in Beeper and find the option to join a Matrix room, then follow the instructions. From memory you might only be able to do this from the desktop application, not on mobile, but the room shows on mobile once you connected on desktop.

Lemmy.NZ has a Matrix chat, see: https://lemmy.nz/post/169187

There is an NZ homeserver: https://mtrx.nz

This is run by fediservices.NZ, the same folks who provide a server for us to run Lemmy.NZ on. But you aren't able to connect a matrix account like this into Beeper, you'll need to use a different app like Element.

Hope this helps!