Removing a restriction site and introducing other at its place
The recognition site for EcoRI is GAATTC, and the enzyme cuts after the first base. See this picture from NEB:



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.