After installation startup X-Plane.
In X-Plane's startup menu place yourself at a busy airport.
Maybe you want to check coverage at the web site of OpenSky Network first: You will at maximum see the same aircraft as show on their page.
No idea? Try EDDF, LSZH, KLAX, or OMDB: They have a pretty good coverage also of ground traffic. To make things perfect choose a gate or ramp position for your own aircraft rather than a runway...you don't want to be in the way of starting live traffic, do you?
Start your "flight"!
Live Traffic should welcome you with a short message on the screen. Message output is generally in the top right corner.
If you run XSquawkBox, X-IvAp or any other multiplayer client in parallel please read about that now.
Switch to Tower View [Shift-5] and maybe from there even to Free-Camera [c] to roam around and watch.
Messages appear in the top right corner informing you about the fact that live data is being requested and that the buffer (hopefully) starts filling up.
As soon as it says "...displaying 1..." there really is a new aircraft in the air...somewhere.
Give LiveTraffic one or two minutes time to collect and buffer data and stabilize derived flight paths, then just enjoy watching traffic for a while.
Click on the Info button in the Aircraft List to open an Aircraft Information window for the particular plane.
If you activate the AUTO option in that A/C Information window, then it will show you information on the nearest aircraft in your camera direction. Very useful when just looking around or our of a cockpit window.
Open X-Plane's internal map: All of LiveTraffic's planes appear there, too.
You'll notice that the aircraft come with yellow labels. You can configure the content of it in the settings. By default it contains a/c type, a unique identifier (flight number, call sign, registration, or transponder hex code, whatever is available first in this order), altitude, and speed. You can type the unique identifier into an A/C Info Window for getting infos on an aircraft of your interest.
At the bottom of the Aircraft Info Window or in the Aircraft List click on Camera...and you hop onto that plane's tail and follow it's movements (see here for more info).
Toggle the Visible checkbox: The plane will disappear and reappear, which you can use to make yourself a slot in a busy queue waiting for take off (see here for more info).
And when you find or make that gap - take off yourself and chase the others...or just link into the chain of approaching aircraft and be a good chain link. Check your TCAS...it might work!