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.
Yes it does increase.
As ndnordic explained Electronegativity is a measure of how strongly the nucleus of an atom attracts electrons to itself.
As you go across the periodic table the amounts of electrons increase and so does the amount of protons. Because of the increase of protons the force of the nucleus is stronger therefore pulling onto the electrons stronger.