The importance of chiral symmetry breaking is a point that e.g. David Gross, a co-father of Quantum Chromodynamics, likes to make whenever some people suggest that the mass is entirely due to the Higgs field. In fact, most of the mass of visible matter is due to the QCD, especially chiral symmetry breaking, and it has nothing to do with the Higgs field.
The Higgs field interacts with leptons and quarks via the "Yukawa coupling" $y\cdot h\bar\psi\psi$ where $y$ is the Yukawa coefficient, $h$ is the Higgs field, and $\psi$ is a fermion field. In the vacuum around us, the Higgs field has a condensate, $\langle h \rangle \sim v$, and this converts the cubic term to the quadratic mass terms for leptons and quarks $m\bar\psi\psi$ where $m\equiv y\cdot v$.
However, the up-quark and down-quark have masses (obtained from the Higgs field and the interaction above) that are equal to 4 MeV and 7 MeV, respectively. If you add three such quarks, you only get 15 MeV or so. However, the mass of the proton is about 938 MeV, almost 1 GeV. So over 98% of the mass of the proton and neutron (over 99% is an overestimate) – and therefore the vast majority of the mass of the nucleons, atoms, and molecules – is due to some additional forces, namely the interactions between the colorful quarks and gluons described by Quantum Chromodynamics.
Chiral symmetry breaking in QCD is a very technical topic. But it is the breaking of the symmetry under a phase redefinition of the quark fields that would act differently on the left-handed 2-component spinors within the quark fields, and differently on the right-handed ones. The vacuum including the violent dynamics of gluons creates another kind of a condensate but it is not made out of null-momentum quanta of the Higgs field; instead, you may imagine that it is made out of quark-antiquark pairs. Unlike the Higgs ocean, this "state of matter" filling the vacuum has a composite structure.
Why the symmetry is broken is a difficult technical issue and so is the analysis of the physical consequences of this symmetry breaking. But it is true that it's responsible for the fact that the proton mass is 50+ times greater than the mass of the three quarks. The origin of the chiral symmetry breaking is the strong nuclear force, mediated by gluons and described by QCD, rather than the weak nuclear force, mediated by the W-bosons and Z-bosons and made possible by the Higgs field. Recall that aside from these two forces, we also have electromagnetism and gravity. At any rate, the forces behind the chiral symmetry breaking and the Higgs mechanism are completely different, non-overlapping. They're described by remotely similar maths but not by the same fields and same interactions.
Leptons don't gain any mass from chiral symmetry breaking because they don't carry the "color charge" and the chiral symmetry breaking only affects particles that interact via the color charge, namely quarks and gluons. However, the electron, the only widespread lepton in the visible matter, is 1836 times lighter than the proton so it represents a tiny minority of the mass of visible matter, too.