How does the Higgs boson give properties to matter such as mass?
I understand that the Higgs boson is one of the subatomic particles that gives matter its properties, but how?
The mechanism that allows the Higgs field to impart mass to matter is spontaneous symmetry breaking. The Mexican hat logo of the Physics Stack Exchange is a visual illustration of this concept. The top of the hat represents the potential energy of the field when the field is zero. It is an unstable equilibrium point that may have occurred a short time after the big bang, but as the universe cooled the field shifted to the brim where the energy is lower. This means that the Higgs field has a nonzero vacuum expectation value. This happens to play the role of mass for any matter field that interacts with the Higgs field. The story of how these ideas came about is convoluted and involves many people who worked during the 1960s-1979s. Together these people were able to unify the weak and electromagnetic interactions.
The Higgs boson is the quantum excitation of the Higgs field. The development of the electro-weak theory led to the experimental search that culminated in the discovery of the Higgs boson in 2013. It has a very large mass and required a very powerful accelerator (the LHC) for its discovery. The Higgs boson only exists briefly in the high temperature debris of elementary particle collisions. The Higgs field and it's nonzero vacuum expectation exist everywhere and imparts mass to matter.