Why does elecronegativity increase across the periodic table?

Electronegativity is a measure of how strongly the nucleus
of an atom attracts electrons to itself.

As you noted,
electronegativity increases as you go across a period on the periodic table. The reason
is as you go from one element to the next you are increasing the number of protons in
the nucleus.  Because the nucleus is small and dense, its positive charge is
concentrated in a small volume. This increasing concentration of charge attracts the
surrounding electrons more and more strongly and thus the electronegativity
increases.

You may have also noticed that electronegativity
decreases down a period. That is because as you go down, the distance from the nucleus
to the surrounding electrons decreases.  Coulomb's law says that the force of attraction
between charges in inversely proportional to the square of the distance between them. So
as the atomic radius increases the attraction between the nucleus and the electrons
decreases quickly.

Coulomb's Law:  F =
kQ1Q2/d^2.

F = force in
Newtons

Q1, Q2 = charge in
coulombs

k = Coulomb's
constant

d = distance between charges in
meters.