I am not much of a networking guy but decided to try the Unifi stuff several years back after some frustration with one of the consumer mesh wifi things. I found the Unifi stuff incredibly confusing when reading about it prior to buying some. Some friends at work who are more networking savvy were very keen on Unifi were very helpful though, and it turns out not to be as strange as I thought, although coming from the garden variety consumer wifi router boxes, it seemed bizarre at the time. I'll try to give an explanation along the lines of what they gave me.
In a wifi router like your TP-Link, the control plane software is running on the box with the the switching hardware and wifi ap, so you've got a little single board computer running web server for the UI, and all the random dhcp/dns/etc and other doodads that can run on them.
In the Unifi world, you've got all the same functions, routing/switching/wifi etc, but instead of sharing one box, the functions are spread across a number of different devices.
As with a combination router/wifi/switch, the important part to a user like you or me is that control plane software- you plug the thing in, point your browser to 192.168.1.1 or whatever, and set things up. The Unifi world has this too, but that software component doesn't need to run in any specific place in your network. So for example, you could buy two Unifi Access Points which do nothing but talk to wifi clients, then you would need some kind of device capable of going your routing, and you might need a switch as well.
Ubiquiti sells a variety of little routers and switches that can perform those network functions, but which don't have any compute or storage resources that would be necessary to run the control plane software. However, they also sell little gizmos like the Cloud Key which can run the control plane software- it's just a tiny server with some flash storage and an ethernet port. I'll refer to that thing as The Controller.
When you change some settings on your TP-Link, the web UI app is twiddling with the the routing/switching/wifi/etc hardware or software on the device. In the Unifi world there's a web ui as well, running on the controller, but when you change a setting in the web ui, the controller decides which devices need to have their configuration updated, and sends out new configuration to them over the network.
Here's where I think things get confusing. The Controller software package can run on a wide variety of devices which are so different that the whole thing will seem nonsensical if you're used to regular wireless routers. You can buy a CloudKey and connect it to your network. You can download a copy of the Controller that will run on a Linux box on your network. Or you could do the same thing but have the Controller running on a machine that isn't on your network at all, like an EC2 vm. Or, Ubiquiti also sell some devices which combine two or more functions into a single device, like some of the "UDM" family of devices have compute and storage resources in addition to the switching/routing/wifi hardware, and have The Controller software installed in advance.
To give you an example of how flexible the controller placement is, I have a little Synology NAS that is able to run Docker, and on it I have an image that contains the Unifi Controller, so when I go the web ui for my network, I'm talking to a containerized web server on the NAS, which is managing the configuration of my devices, which are a router (I just replaced my USG 3P with a UXG-Lite yesterday), a couple of their little inexpensive switches, and a pair of Wireless Access Points.
What I like about this model is that I'm able to update pieces of it as I need to, and usually the individual pieces are fairly inexpensive. But what I dislike about it, as some other respondents on this thread have complained, choosing which devices you need is confusing as hell. They sell at least one machine which has a wifi AP, switch, router, and controller all in one box. Why not get that? The reason, I believe, is that many of the people who use this Unifi stuff are managing a bunch of networks at a bunch of different sites, like maybe at a bunch of retail locations or restaurants, where its way more convenient to have the controller running offsite, but then they decide to install some stuff at home and need a Controller which needs fewer resources so a Cloud Key or just running the stuff on your desktop would be ok.
This flexibility means that there's no single right set of hardware, no single best product, etc. Especially if you don't need multiple APs, I think a single-box wifi router will provide equal or superior performance with much less trouble, but once you need multiple APs, the Unifi stuff can be compelling if you're comfortable with the architecture, but I think it's difficult to decide which hardware bits to choose and what the best place to run the controller will be.
anyway apologies for the length- I found all this very confusing initially as well although I've grown fond of the Unifi stuff and thought it might be worth writing the whole thing out in case its useful to somebody considering this stuff.