The Higgs field is a quantum field that permeates all of space. It interacts with particles and gives them mass. However, it's important to note that the Higgs field does not only give mass to moving particles. The interaction between particles and the Higgs field gives rise to their mass, regardless of whether they are stationary or in motion. Even particles at rest have mass due to their interaction with the Higgs field.
The increase in mass for particles moving near the speed of light is explained by the theory of relativity, specifically by the concept of relativistic mass. According to Einstein's theory of special relativity, as an object moves faster and approaches the speed of light, its relativistic mass increases. This effect is not directly related to the interaction with the Higgs field.
In the context of the Higgs mechanism, the Higgs field gives mass to particles, but it doesn't directly depend on their velocity. The mass of a particle is determined by its interaction with the Higgs field, which is a fundamental property of the particle itself, independent of its velocity. So, while the Higgs field is responsible for giving particles mass, it doesn't explain the specific increase in mass with speed predicted by relativity.