Why don't primates have galactose-alpha-1,3-galactose (alpha-gal), but other mammals do?
The general idea is that infectious agents use certain cell-surface proteins to enter the cell, and presumably recognize them by their particular glycoproteins. There are examples (cited in the extract below) of changes in e.g. blood group antigens in populations that are thought to convey (at least temporarily) an advantage against particular pathogens.

In this context, let me cite the rest of the section of the article that you didn’t have access to: 


  However, after the geographic separation of Africa and South America,
  Old World primates appear to have been subjected to a selective
  pressure for the suppression of autologous α-galactosyl epitope
  synthesis and for the subsequent production of anti-Gal. Possibly, an
  infectious agent endemic to the Old World, which was detrimental to
  primates, and which expressed α-galactosyl, or α-galactosyl like
  epitopes, exerted a selective pressure for the evolution of Old World
  primates able to suppress autologous α-galactosyl epitope expression.
  Thus, immune tolerance toward this epitope was lost and anti-Gal
  antibodies could be produced in these primates as a protective
  response to such putative pathogens. 
  
  An alternative scenario for the
  evolutionary suppression of α-galactosyl epitope expression could
  have been one in which ancestral Old World primates suppressed
  expression of these epitopes on their cell surface glycoconjugates as
  a result of exposure to a pathogen which exerted its detrimental
  effect via α-galactosyl epitopes functioning as receptors. A current
  example for such a pathogen is Clostridium difficile. Enterotoxin A
  produced by this pathogen was found to bind to the cells it damages
  through interaction with α-galactosyl epitopes on cell surface
  glycoconjugates [12, 56]. A bacterial toxin with a similar ligand
  specificity and which was detrimental to ancestral Old World primates
  could exert a selective pressure for the evolution of primates which
  suppressed the expression of α-galactosyl epitopes and, thus, lost
  the immune tolerance toward this carbohydrate structure.
  
  [12] Clark GF, Krivan HC, Wilkins TD, Smith DF (1986) Toxin A from Clostridium difficile binds to rabbit erythrocyte glycolipids with terminal  Galα-3Galβ1-4GlcNAc sequences. Arch Biochem Biophys 257:217
  [56] Krivan HC, Clark GF, Smith DF, Wilkins DT (1986) Cell surface binding site for Clostridium difficile enterotoxin: evidence of a glycoconjugate containing the sequence Galα→3Galβ1→4GlcNAc. Infect Immun 53:573


Whatever the reason, the consequence of this was:


  This also resulted in the loss of immune tolerance to the α-galactosyl epitope and the production of anti-Gal. 


I presume this explains the IgE problem referred to in a comment, and I would consider this as an unfortunate side-effect of the change, and not a reason for it.