How do the three mechanisms of the 1000X improved "new" vancomycin work together?
This is typically for a paper under embargo - it can be accessed by registered journalists so they can cover the original article when it is published. The paper is now published, so I changed the first answer.

Before I go into the details, it is necessary to have a short look at the working mechanism of Vancomycin and the way bacteria got around it (which took exceptionally long given that the drug is used since 1959 in the clinic):

Vancomycin works by inhibiting the proper crosslinking of the bacterial peptidoglycan in the cell wall. This is achieved by binding to the substrate for the peptido glycane synthesis, a D-alanyl-D-alanine end of a peptide. Cells then burst due to osmotic pressure.

Resistant cells have a D-alanyl-D-lactate ending of this peptide (all details can be found here or in the introduction of the paper) which lowers the effictivity of the drug by a factor of 1000. Interestingly, this resistance mechanism seems not to have evolved in pathogen bacteria itself, but has been "borrowed" from bacteria which produce Vancomycin and need this as a self-defense (for more, see reference 2).

Vancomycin forms a pocket around the d-Ala-d-Ala substrate, which is impaired in the resistant form. Overcoming the resistance means finding a variation of Vancomycin which effectively binds both of the variants and thus inhibits proper cell wall crosslinking.

To achieve this, Okano and his colleagues chemically modified the structure of Vancomycin with several different functional groups at different sites of the molecule, see this figure from the paper (figure 4 from reference 1):



Changing these residues changes the sensitivity against these modified Vancomycine quite a lot, as can be seen in the sensitivity table in the figure. 



When they changed the molecule, they did not only change the binding pocket to allow binding to the different peptide substrates, they also introduced two other independent mechanisms against the bacteria. The most active compound (named 18 in the paper, seen above from figure 11) shows:


  Compound 18, which incorporates the redesigned pocket modification for
  dual d-Ala-d-Ala/d-Ala-d-Lac binding (blocks cell wall synthesis by
  ligand binding, including inhibition of transpeptidase-catalyzed
  cross-linking), the CBP disaccharide modification (blocks cell wall
  synthesis by direct transglycosylase inhibition without
  d-Ala-d-Ala/d-Ala-d-Lac binding), and the C1 quaternary ammonium salt
  C-terminal modification (induces membrane permeability), exhibited the
  most potent inhibition of cell wall synthesis in the assay of all
  compounds assessed.


Okano and colleagues also tested the different modifications one by one to see, which one has which effect and also tested the effectivity on resistant bacteria. The resulting table shows that the strongest effect is only achieved when all three modifications are present in the molecule. 



Since these are independent mechanisms, it will also prolong the time until resistance against the new version of Vancomycin (given that is has a tolerable toxicity in humans and can be introduced into the clinic) will take a while to develop. Despite the optimism in parts of the paper this resistance will eventually develop, although it is quite complicated to overcome three different mechanisms.

References: 


Peripheral modifications of [Ψ[CH2NH]Tpg4]vancomycin with added
synergistic mechanisms of action provide durable and potent
antibiotics
Glycopeptide antibiotic resistance genes in glycopeptide-producing
organisms.