If you want to record audio onto your computer, you need to have a way to connect a microphone to that computer. An audio recording interface is what allows you make that connection.
However, it's much more than just physically connecting a mic to a computer. What really matters is what the interface does.
A microphone is able to create an electrical signal out of sound. Depending on the kind of mic you are using, this is done in a couple of different ways. See my article Condenser vs Dynamic Microphone: What Is The Difference? for more on that.
But just having an electrical signal isn't enough. A computer needs two things:
When using a standard computer microphone - the kind that has a 3.5mm pin that connects directly to a computer's built-in sound card - you have to put both preamp and converter into a very small space on the motherboard. THAT serves as the "interface." So these tend to be of pretty poor quality.
Another way to connect a microphone to a computer is with a USB mic. In that case, the preamp and converter are built into the microphone itself. The allows more space and higher quality components. That's why many USB microphones are able to record at pretty high quality levels.
But you can get the highest audio quality (in most cases) with an interface unit that goes between a standard microphone (connector has 3 pins called an XLR connector) and the computer.
These interface units are usually some kind of box that hooks up to the computer with a USB cable. These boxes have enough space to put in very high quality preamps and converters. They have XLR connectors on them where you can plug your microphone in.
Once everything is hooked up - mic to interface, then interface to computer - you can control the input level from a knob on the interface to record just the right level.
The USB connection on the interface also receives sound from the computer. You can connect speakers to the interface, as well as headphones, to hear playback. There is also usually a way to monitor what you are recording through your headphones, which allows you to have "zero-latency" monitoring. That means you don't have to wait for the audio to go into the computer and then back out again to hear your voice (if recording vocals) WHILE your are recording.
If you did have to wait for that, there would be a delay between when you speak, and when you hear your words. That is called "latency." So it is really helpful to have the zero-latency monitoring capability that recording interfaces usually offer.
One other thing most interfaces provide is something called phantom power. Certain microphones, called condenser microphones, require phantom power. So if you are using one of these mics, you just need to turn on the Phantom Power switch. Just make sure you turn phantom power off if you use a dynamic mic. For a review of the difference between condenser and dynamic microphones, see my article Condenser vs Dynamic Microphone: What Is The Difference?
With very few exceptions, you can achieve the best audio quality using a good standard (NOT USB) microphone connected to an audio interface. So if your budget allows for it, I would say yes, you need an audio interface if you want the highest audio quality. There are some very good USB microphones out there. And if your budget is limited, those are the best bang for the buck.
Like so many things, there are tons of choices out there. And part of the decision will depend on on how many microphones you need to record simultaneously. If it is only ever going to be one microphone at a time, you can get a very good quality interface for around $100.
One very popular choice is the Focusrite Scarlett series of interfaces. The Scarlett Solo has one microphone input and costs about $110.
Focusrite makes several other interfaces as well, with more mic inputs and other capabilities.
A few other popular interfaces include the Tascam US-1x2, PreSonus Audiobox, Solid State Logic SSL2 interface and MOTU M2 interface. These are all under $300. There are, of course, many more models of different shapes and sizes. But one of those should be good enough for home recording studios.