Could someone smarter than me explain Matrix to me?

I wouldn't assume that I'm smarter, but I do have more than a little experience here, so I'll try to answer your questions. :)

It's a real-time messaging platform. The most common use for it is text chat, both in groups (like Discord or IRC) and person-to-person (like mobile phone text/SMS). It supports other uses as well, like voice chat, video conference, and screen sharing, although much of that is newer and gradually showing up in clients.

What would be the utility for someone, who cares about privacy and currently uses Signal and email for communication?

Compared to Signal:

• Matrix doesn't require a phone number, or even an email address (although some public homeservers want an email address these days, as a recovery method in case you forget your password).
• Matrix has a variety of clients, so it's more likely that an app fitting your needs exists.
• Matrix clients typically don't require Google services at all; neither to get the software nor to receive notifications.
• Matrix cannot be monitored at any single location, so it's more resistant to meta-data tracking at the network level.
• Matrix cannot be shut down by any single organization, so it's more resistant to censorship and denial-of-service attacks. If a homeserver is ever forced offline, only the accounts on that homeserver go away; all your other contacts remain intact. Same thing if a service operator changes its policies or goes out of business.
• Matrix (last time I checked) had better support for using multiple devices on the same account. Phone, laptop, and office computer, for example.
• Matrix homeservers can be self-hosted by anyone, and still participate in the global network.
• Signal's encryption covers more meta-data at the application level than Matrix currently does. This might be important if you're a whistleblower or journalist whose safety depends on hiding your contacts from well-positioned adversaries.

Compared to email:

• Matrix has end-to-end encryption, with forward secrecy, built in. It's generally better for privacy than bolting PGP onto email, and it's far easier.
• Matrix is well suited to instant messaging.
• Matrix supports features that people have come to expect from modern chat platforms, like reaction emoji and editing after sending.
• Email has a greater variety of servers and clients.
• Email apps often have more composition features to support long-form writing.

What advantage would it give me over other services?

We already covered Signal, and there are too many other services to compare every difference in all of them, but here are some more common advantages:

• Matrix is a completely open protocol, developed through a public and open process, with open-source servers and client apps. This is important to people who care about privacy because it can be scrutinized by anyone to verify that it operates as it claims to, and can be improved by anyone with a good idea and motivation to participate. It's important to people who care about longevity because nobody can take it away.
• Matrix has multiple clients for every major platform: desktop, mobile, and web.
• Matrix handles groups of practically any size (including just one or two people).
• Matrix messages are delivered even when you're offline.

Is Matrix anything good already, or is it something with potential that’s still fully in development?

Until recently: Ever since cross-signing and encryption-by-default arrived a couple years ago, it has been somewhere between "still rough" and "pretty good", depending on one's needs and habits. I have been using it with friends and small groups for about five years, and although encrypted chats have sometimes been temperamental, they have worked pretty well most of the time. When frustrating glitches have turned up, we sorted them out and continued to use it. This has been worthwhile because Matrix offers a combination of features that is important to us and doesn't exist anywhere else. I haven't recommended it to extended family members yet, because not everyone cares as much about privacy or has the patience for troubleshooting in order to get it. However...

Recently: The frequency of glitches has dropped dramatically. Most of the encryption errors have disappeared, and the remaining ones look likely to be solved by the "Invisible Encryption" measures in Matrix 2.0. Likewise with things like sign-in lag and client set-up.

If you're considering whether it's time to try it, I suggest waiting until Matrix 2.0 features are formally released in the clients and servers you want to use, which should be very soon for the official ones. I wouldn't be surprised if I could confidently recommend it to family members in the coming year.

How tech savvy does one need to be to use Matrix?

If you just want to chat, not very. Even one or two of my friends who can barely use email got up and running pretty quickly with a little guidance. Someone who can get started using Lemmy by themselves can probably handle it on their own.

If you want to host your own server, moderately tech savvy.