Question about the DNA glycosylase enzyme assay/formamide gel loading buffer
Escherichia coli endonuclease III (which I will now refer to as Nth), and the human homolog hNTH1, has both glycosylase and AP lyase activity. The section of the article you're referring is describing an assay of lyase activity. The assay tested both Nth and hNTH1 as well as HAP1, a human AP hydrolase that cleaves 5' to the AP site. Nth and hNTH1 were found by this assay to cleave 3' to the AP site. HAP1 is dependent on $Mg^{2+}$ for catalytic activity. hNTH1 (and likely Nth, given the high conservation) are dependent on $Mg^{2+}$ for substrate specificity (Eide et al., 2001). hNTH1 and Nth contain an $Fe-S$ cluster that is involved in DNA binding (Thayer et al., 1995).
Originally I thought that the reaction is stopped with EDTA. EDTA is a chelating agent; it binds and sequesters metal ions and prevents their use by enzymes. EDTA would prevent catalysis by HAP1 and DNA binding by Nth/hTNH1.
However, Chris pointed out that the loading buffer contains 1 mM EDTA while the reaction buffer contains 5 mM $MgCl_2$. While the paper doesn't give the volumes of reaction and loading buffer mixed together, it is perhaps unlikely that EDTA chelation of $Mg^{2+}$ is the sole mechanism for stopping the reaction. The high concentration of formamide (80%) in the loading buffer could then be responsible. Formamide is commonly used to denature nucleic acids and, at high concentrations, would likely have an effect on enzyme function.
References

Eide L, Luna L, Gustad EC, Henderson PT, Essigmann JM, Demple B, Seeberg E. 2001. Human Endonuclease III Acts Preferentially on DNA Damage Opposite Guanine Residues in DNA. Biochemistry-US. 40(22):6653–6659
Thayer MM, Ahern H, Xing D, Cunningham RP, Tainer JA. 1995. Novel DNA binding motifs in the DNA repair enzyme endonuclease III crystal structure. EMBO J. 14(16):4108–4120.