The Boussinesq equation (BqE) has been of considerable interest in coastal and ocean engineering models for simulating surface water waves in shallow seas and harbors, tsunami wave propagation, wave over-topping, inundation, and near-shore wave process in which nonlinearity and dispersion effects are taken into consideration. The study deals with the dynamics of localized waves and their interaction solutions to a dimensionally reduced (2 + 1)-dimensional BqE from N-soliton solutions with the use of Hirota’s bilinear method (HBM). Taking the long-wave limit approach in coordination with some constraint parameters in the N-soliton solutions, the localized waves (i.e., soliton, breather, lump, and rogue waves) and their interaction solutions are constructed. The interaction solutions can be obtained among localized waves, such as (i) one breather or one lump from the two solitons, (ii) one stripe and one breather, and one stripe and one lump from the three solitons, and (iii) two stripes and one breather, one lump and one periodic breather, two stripes and one lump, two breathers, and two lumps from the four solitons. It is to be found that all interactions among the solitons are elastic. The energy, phase shift, shape, and propagation direction of these localized waves and their interaction solutions can be influenced and controlled by the involved constraint parameters. The dynamical characteristics of these localized waves and their interaction solutions are demonstrated through some 3D and density graphs. The outcomes achieved in this study can be used to illustrate the wave interaction phenomena in shallow water.