One important factor influencing the rate of chemical reactions is the concentration of reactants. The higher the concentration of reactants, the more frequently the reactants bump into each other giving them the opportunity to react. One way to speed up a chemical reaction (other than increasing the concentration of reactants) is to tether the reactants in order to increase their effective concentration (for example, see David Liu's work on DNA-directed synthesis, which uses DNA base pairing to bring small molecules together for chemical reactions). In the case of the tosyl chemistry described by Tsukiji et al., tethering the reactive group to the protein will greatly increase the frequency with which the reactive group collides with amino acids on the target protein surface, which in turn greatly increases the reaction rate. This is the "proximity effect" to which the authors refer.
The key to the paper seems to be that the tosyl reagents the authors produce are very poorly reactive. They report labeling yields of 75% when they react 40ÁM of target protein with 80ÁM reagent for 48 hours. In comparison, typical protein labeling reagents (such as N-hydroxysuccinimide or maleimide reagents) will give >90% labeling yields in a few hours. Thus, the reagent seems to be so poorly reactive that it will not react unless its reactant is present at very high concentrations (achieved here through tethering).
Now, the reagent is not 100% specific. If you look closely at the fluorescence image in figure 2b, you can see faint signal from the other proteins in lanes 1 and 2. So, the reagent is capable of reacting with any nucleophilic amino acid residue. However, these reactions occur very slowly unless the reagent gets tethered to a protein, in which case the reaction will occur at a much more rapid rate (although the labeling rate is still very slow compared to typical labeling reactions).
The slow rate of reaction is likely to limit the applicability of this technology and why I would not bet on this technology replacing GFP. Whereas GFP fluorescence matures in the tens of minutes, the tosyl chemistry seems to require a day or more of incubation with the ligand to achieve quantitative labeling. This can be problematic as may proteins turnover on a much faster timescale.
To combine this with some thoughts from another one of your posts, perhaps the ligand-dependent tethering idea could be used to speed the rate with which click chemistry reagents react with their targets.