It just tells you what classes are available, how many students are in each class, and what lectures/tutorials/labs/other classes you need to take. It's just like the Handbook version (e.g.
http://www.timetable.unsw.edu.au/2013/MATH1141.html ) however, it updates more frequently (from what I've seen).
So what you do is you decide which subjects you want to take, so let's say MATH1141. You go to Class Util and find "MATH". Select the appropriate semester, then find "MATH1141". It shows you two lecture streams, both which you need to take, and it shows you TUT1 and TUT2, i.e. you choose 2 tutorial streams. If we choose the first option, it says "Fri 14 (w2-4,5-13, MechEng403)". This means your tutorial will run on Friday, it starts at 14:00 or 2pm and runs for 1 hour unless it specifies a time range, it will run from weeks 2 to 13, and your room is Mechanical Engineering Building, Level 4, Room 403.
You can go to the Handbook version for a slightly better interface, but the Class Util is best to see how many spots are available and whether a class may open up.