Is matter a continuous part of the field of space-time? I recently found this quote by Einstein (in On the Generalized Theory of Gravitation, 1950), and it seems to me like he is saying that matter is a part of the field of space-time, and is nothing more than an area of the field with very high energy. Is this correct?

According to general relativity, the concept of space detached from any physical content does not exist. The physical reality of space is represented by a field whose components are continuous functions of four independent variables-the coordinates of space and time. It is just this particular kind of dependence that expresses the spatial character of physical reality. Since the theory of general relativity implies the representation of physical reality by a continuous field, the concept of particles or material points cannot play a fundamental part, nor can the concept of motion. The particle can only appear as a limited region in space in which the field strength or the energy density are particularly high.

 A: This quote is within the framework of General Relativity only. It accepts that there exist "particles"  but the only materialization of their existence as particles appears in the change of the functional form of  the four independent variables-the coordinates of space and time. In such a mathematical model of Nature continuity is built in.
Now the question is how well this theory/model agrees with experimental observation and astrophysical ones. In astrophysics it does very well, it has not been falsified. ( note a theoretical proposal cannot be proven true, it can be validated by observations and falsified if one observation disagrees with predictions). So if we consider matter as just clusters of galaxies, galaxies, stars, planets and even satellites one can give an A++ for the validity of the General Relativity predictions and continuity of space time functions is one of them.
This is not true when we enter the realm of atoms, molecules and  elementary particles . At the micro level quantum mechanics reigns and all our experiments point out that they are discontinuous entities with mass and  discrete energy levels and discrete interactions. Three more interactions in addition to  the gravitational one, which last General Relativity models as space time formation. Thus the quote is falsified by experimental data at the microscopic level.
At the moment the three elementary interactions, strong, weak and electromagnetic are unified in a theory called the Standard Model . Just today we learned that the Nobel prize for physics was awarded to prof Higgs and Englert for predicting this completion of the Standard Model decades ago, with the discovery of the Higgs by CERN groups in 2012. 
The goal of theoretical physics is to unite all forces , including gravitation in one theoretical framework that will give the Standard Model at microcosm and General Relativity for the cosmos. Many people are working on this. At the moment a dominant theory is String Theory which can embed the Standard Model of particle physics which encapsulates all  experimental data , and also quantize gravity , a theoretical problem that has been hard to tackle the last 60 or so years. It is work in progress and many theorists are working on this.
The ideal theory then will have as an approximation for large masses and distances the quote above, while for the microscom will retain the observed discontinuities of the real masses of real particles in space time.
