Proper Lorentz Transformations Im studying the Minkowski 2d spacetime.
Why is a proper lorentz trasformation so important? And what the intuitive meaning to say that a proper lorentz trasformation "can continuosly deformed to an identity"?
 A: This question could just as easily apply to rotations, so I'll explain it in that context first.
A proper rotation is a rotation which can be built from the identity (i.e. no rotation at all) by making a sequence of tiny rotations.  If I want to rotate my coffee cup by $180^\circ$, I can simply perform 180 tiny rotations in which I rotate my cup by $1^\circ$ each time.  In general, a $180^\circ$ rotation can be written as $N$ rotations of $\left(\frac{180}{N}\right)^\circ$ each.
In the limit as $N\rightarrow \infty$, I am performing an infinite number of infinitesimal rotations to build up my full rotation.  The fact that I can do this is useful in part because it's often mathematically easier to deal with infinitesimal transformations.  This is what is meant by the "continuous deformation" you mention in your question.
An improper rotation, on the other hand, can not be built in this way.  An example of an improper rotation is a reflection.  If I write my name on my coffee cup, no amount of rotating around any axis will ever make the letters appear backward.
So, the rotation group (the set of all transformations which can be considered rotations) is split into two distinct and disconnected subgroups - the proper rotations, which can be continuously built out of infinitesimal rotations starting from the identity, and the improper rotations, which cannot.
Lorentz transformations include rotations as well as boosts, but the same idea applies.  If a Lorentz transformation can be built from the identity by a sequence of infinitesimal boosts and/or proper rotations, it is a proper Lorentz transformation.  If it requires a spatial reflection and/or time reversal, it is called an improper transformation.

As far as why proper Lorentz transformations are important, the fundamental idea is that they encode a fundamental symmetry of the laws of nature.  As far as we are aware, all of the laws of physics are invariant under proper Lorentz transformations.  That is to say, if two observers occupy two different reference frames which are related to each other by a proper Lorentz transformation, then the laws of physics should be the same for each of them.  This is called the equivalence principle.
Note that this is not mean that they should observe precisely the same thing.  Obviously if two observers are watching a ball fly through the air and one is rotated with respect to the other, then the coordinates of the ball as a function of time will be different for each.  However, they should both see the ball obeying the same "rules."
The same is not necessarily true of improper Lorentz transformations.  The second law of thermodynamics is not symmetric under time reversal, and the weak interaction is not symmetric under spatial reflections.  Some laws of nature have these additional symmetries, but they are not implied by the equivalence principle.
