An audio interface connects external sound sources (e.g. microphones, guitars etc.) to your computer – it converts analogue signals into digital information that your computer can understand. Conversely, it also converts the information inside your computer back into electrical currents that your speakers (or headphones) can understand.
Most computers come with some sort of on-board audio interface (or “sound card”), but these are generally poor quality and offer extremely limited types of connectivity (i.e. there’s nowhere to plug your guitar in!).
If you take your music seriously, you will care about ensuring that the sound that goes into and comes out of your computer is not coloured in any way by the process. A good audio interface will provide a clean signal path which will ensure that this is the case.