If you think that those 85 pages are too long, the main resulting formulae are (3.18) on page 18 with (3.14) on page 17 that you need to know:
The path-ordered exponential, the exponential prefactor with "L_0" in it, and the choice of contours are things that, I believe, no person with IQ below 250 could guess without lengthy calculations. :-) Nevertheless, let me sketch why there are the two integrals. The closed string state is composed out of different points on the closed string - the closed contour integral - and each of these point contributions is able to link the point to any other point on the closed string - the open contour integral in P. The linking is given by open string evolution (the path-ordered exponential).
Amazingly enough, they show that in the Schnabl gauge, these formulae can be computed in analytic form and the resulting boundary states agree with the states you would expect for several known string field theory solutions.
At any rate, the paper improves the people's ability to directly translate between different descriptions of configurations of D-branes and their generalizations. They also increase our understanding of closed strings in open string field theory. These direct dictionaries are very important because they reduce the mystery why there are so many ways - choices of degrees of freedom - that describe the same stringy physics. Once you know the "field redefinition", the two descriptions cease to be "genuinely different".
All their formulae are "tangible" i.e. more than formal: they can be expanded in terms of string oscillators and all the coefficients are finite, unlike the case of some old misleading heuristic Ansätze.
I would also say that the second best paper today is the second paper ;-) by Rolf Schimmrigk, "Applied String Theory", although it is far less technical (perhaps atypically for this mirror symmetry expert). He says that there is nothing new or bad about string theory's having many solutions: it's been known for decades and a good theory must describe all possible universes, not just ours.
Much like in previous cases of physics, experimental data are helpful to choose the correct vacuum and he explicitly mentions cosmological, gravitational, and other experimental constraints and future observations that will help to locate the right vacuum (or their class).