It is easy to understand in a simple local U(1) symmetry case. SU(2) case is a bit more difficult.

The Higgs potential V(φ)  develops a classical minimum away from φ=0. Quantum fluctuations about this classical minimum are denoted by η (say). After doing some easy algebra one finds that  η field is massive and m2η = 2v2λ. Here λ is the Higgs quartic coupling and v is VEV. See Halzen and Martin Eqns. 15.54 and 15.55

Eventually a main physics goal of HL-LHC experiment is to probe into Higgs quartic coupling or the Higgs self coupling.