The argument is circular: electronegativity is a measure of the tendency of atoms to attract electrons. Therefore electrons are attracted more strongly to electronegative atoms. More attractive has no more meaning than "bound more strongly" and a measure of binding strength can be regarded as a mixture of ionization energy and electron affinity, that is, an electronegativity parameter as defined by Mulliken.
Pauling's original definition of electronegativity relied on the idea that the bond dissociation energy contains contributions from a QM interaction plus an additional purely electrostatic interaction (a classical Coulombic interaction not involving a QM description of chemical bonds). In homonuclear diatomic molecules the electrostatic term is not important because the atoms "tug" at the electrons with equal strength, and so only a QM term contributes to bonding. For heteronuclear diatomics one can estimate the magnitude of the contribution of the QM term as an average of the terms in the two homonuclear molecules. Subtracting this average from the dissociation energy for the heteronuclear diatomic then results in an estimate of an electrostatic contribution to bonding. The electrostatic contribution is argued to be due to the different electron attraction in the two atoms, which is encoded in Pauling's electronegativity parameter. This is summarized by the equation for the heat of formation postulated by Pauling (in $\pu{kJ/mol}$):
$$Q_f = 100 (\chi_A-\chi_B)^2$$
The additional electrostatic term enhances the binding energy:
The greater the separation of two elements on the electronegativity scale, the greater is the strength of the bond between them. The extra stability is the resonance energy between the normal covalent structure and the ionic structure.
The excerpt comes "from the horses mouth" (Pauling's General Chemistry textbook). If the explanation is not satisfying I recommend looking into more advanced theories.
