The recognition site for EcoRI is `GAATTC`, and the enzyme cuts after the first base. See this picture from NEB:
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The overhang is: AATT, which is supplied by your oligo and fits into the overhang. If the next nucleotide whould be a `C` than the site would be recreated, since its a `G` its not. The new sequence of the old EcoRI site is `GGATTG` which is not recognized by EcoRI.
One technical note: To avoid religation of the digested DNA which carries this site, I would treat it with alkaline phosphatase to remove the phosphate end at the 5' end of the DNA.