How does electronegativity change from left to right?

Electronegativity is conceived to be the ability of an atom in a chemical bond to polarize electron density towards itself. There are various scales, of which the Pauling scale was the earliest and still most widely used.

If we discount the Noble Gases, we can say that Electronegativity increases across the Period from left to right, and decreases DOWN the Group.

Two factors are important here: (i) nuclear charge; and (ii) shielding by other electrons. Across the Period (same direction) nuclear charge wins in that incomplete electron shells shield nuclear charge very imperfectly. Thus fluorine and oxygen are the most electronegative atoms of VI and Group VII respectively, though fluorine is more electronegative than oxygen. The best manifestation of this is the well known decrease in atomic radii across the Period.

Down a Group, nuclear charge is shielded somewhat by complete inner electron shells. In addition the attraction between the nucleus and the valence electrons is vitiated by the increasing distance between the parties. Fluorine is more electronegative than chlorine, which is more electronegative than bromine.....

And thus across the Period, nuclear charge is the determinant.