In the SM we can not add fermionic mass terms like $m \overline{e}_R e_L$ to the Lagrangian since these terms are not invariant under $SU(2)\times U(1)_Y$.
After introducing the Higgs in the unitary gauge $\phi=\left(\begin{array}{c} 0\\ v + H(x)\\ \end{array}\right)$ into the system we break the symmetry and are able to put mass terms into the Lagrangian which look like $$y \overline{\Psi}_L\phi e_R$$ with $\Psi_L$ the doublet left handed fermion field and $e_R$ the singlet right-handed fermion and the Yukawa coupling $y$. They will provide us mass terms like $m_e \overline{e}_L e_R$ with $m_e$ dependent on the vev and the Yukawa coupling.
Why do we need these $SU(2)\times U(1)_Y$-invariant terms if our symmetry is already broken? How is this method better than just putting the mass terms $m \overline{e}_R e_L$ in our system by hand neglecting the $SU(2)\times U(1)_Y$ symmetry?