Why are we on such a topic anyways?

Zabuza already explained how the Sharingan works in regards to copying jutsus. The most basic explanation is that the Sharingan is an expert at observing details(
link)
The Sharingan can see a multitude of things:
1- it notices minuscule movements such as handseals(
link) and can recognize and memorize muscular sequences and patterns
2- it sees chakra(
link) as well as attribute color to chakra(
link)
3- it can see chakra flow(
link)
Now ability
1 is the basis for the ability to predict movements(speaks for itself I believe). In a same way, in conjunction with abilities
2 and
3, the ability to copy jutsus is reached.
In fact, as Zabuza stated, it's not so much an
ability as it is a sequence of actions & abilities which mesh together to form the action of copying

.
When an enemy uses a jutsu, the Sharingan will dissect, decode and analyze the event. It will notice movements such as hand seals, changes or shifts in chakra nature as well as directional flows of chakra.
Now, using this information, the Sharingan user has all the theoretical information needed to actually understand the basics and use said jutsu. Now, as others have pointed out, the Sharingan user must be adept enough to use the jutsu the Sharingan read.
If however the Sharingan user meets all requirements(such as skill level, elemental affinity, proper chakra control etc.), then he/she can use the jutsu and effectively "copy" it.
In Kakashi's case, he takes it a step further and goes into mind games. As Zabuza explained, he copies a jutsu and instills doubt within his opponent. Then, he continues copying, but actually uses the Sharingan's ability to predict movement so he can keep up with a hand seal sequence
as it is being done(rather than doing it after). Finally, Kakashi uses a genjutsu to signal the opponent into what jutsu he wishes them to use and uses it before them, making the opponent believe that he "saw the future".
Kakashi's copying trick is actually a very well thought and elaborate set of mind tricks in conjunction with the Sharingan's abilities
But really, this is more pertaining to the Sharingan itself than Sasuke.