I can't say for certain whether the Focusrite Scarlet comes with the USB cable, but my guess is yes. Looking at the picture of the back of the unit, it uses a standard USB cable.
The two channels of audio being recorded are sent to the computer over the USB cable. Also any tracks you have previously recorded are sent from the computer back to the audio interface so you can monitor that audio using your headphones and sync your current performance to what was previously recorded (this is known as overdubbing).
These units (audio interfaces) are indeed designed to connect to computers for recording audio into the computer. This product is a consumer version of what is used in professional recording studios, costing many thousands of dollars. In my case I can record up to 16 channels (tracks) of audio at the same time, while also listening to what was previously recorded. Truly professional setups can record more tracks than that.
I would think the Scarlet is pretty much plug and play (unlike my Digidesign interface). The learning curve comes with mastering the recording software (ProTools, for example). It is pretty simple to record and play back audio, but if you want to get into editing the audio (pitch correction, lining up audio to bars and beats, applying different effects to the audio like reverb, chorus, etc.), then it takes some reading of the software manual and playing around with it.
Like I said above, if you want to get started with recording audio to your computer, an entry level unit like the Scarlet is the way to go. There are of course, other interfaces. Visit Sweetwater.com and search for audio interfaces.