[As an analogy] You have to think of Higgs as iron balls that can move on strings like beads. Whereever they go, the strings become heavier and the tiny little balls that can be there gain the weight of this iron ball. In other words only when the iron ball stretches the string with its heavy mass anything else also gain that heaviness. Higgs can give mass to other particles because its a massive particle moving through the “strings” [paths] of spacetime.
Higgs is said to be giving mass to other elementary particles in the world because of the scheme in which elementary particles share their energy and mass among each other. In other words every particle is giving mass to every other particle, but Higgs which is a special particle was missing so far. Most likely they have discovered it at CERN experiment. But every particle gives every other mass simply because they exert force on each other [or share energy with each other equivalently]
This confusion comes because mass of particles are a constant quantity [rest mass]. Hence if a particular “guy” is missing from a sports and you have planned everything wrt that guy having to deliver some important aspects then all the other aspects will be impacted. eg If Mr Rakesh didn’t give milk to everyone in town they will go hungry. But everyone might still get their milk through other possible ways. It would be misleading to say Higgs is the guy who gives mass to the particle in the sense that if it does not give mass the particles won’t get the mass. Higgs ALSO gives mass to other particles is the right way to say it. If Higgs won’t give mass the scheme would break and you have to look for fixing it. Much like if Rakesh is sick find another person or go without milk for sometime. Neither is the Lagrangian represented as a Higgs Lagrangian through many images is any special. In-fact its a general Lagrangian not anything to do with just Higgs unless Higgs purported properties are also implemented.
And the mechanism of spontaneous symmetry breaking is properties of not just Higgs, its a general property. When the nature of a force changes the nature of underlying symmetry change, which through an association of equation of motion with various groups, such as SU(2) reflects this fact.