There are many common explanations of how the Higgs effect gives particles mass, and virtually all of them are either misleading or wrong.
One well-known explanation likens the Higgs effect to having space like a room full of people in a party, and when a massive particle enters that space it's like a famous person - everyone clusters around that person making it hard for that famous person to cross the room, so that fame gives people a form of inertia. This is wrong.
Other explanations imply that the Higgs effect somehow slows particles down like a marble falling through honey as against falling through air. This is also wrong.
How the Higgs effect gives particles mass is very easy to understand, and it doesn't involve any stickiness or clustering or slowing down.
To understand we have to use Einstein's equation e = mc2 - mass is related to energy: the more energy, the more mass.
Now, imagine you have a bar magnet, one end North, the other end South. If you allow it to move freely, it will end up aligned with the Earth's magnetic field, the stable lowest-energy state. The bar magnet will have a mass, which will include the energy of the interaction with the Earth's magnetic field. Now, turn the magnet 180 degrees and hold it there. The magnet is now in a higher energy state, with its field opposite to that of the Earth's. This higher energy means that the bar magnet will have more mass: it will weigh more and it will have more inertia and a higher (but very, very tiny) gravitation field. Changing the energy of interaction of the bar magnet with the Earth's magnetic field changes the mass of the bar magnet.
Magnetic fields, like many other fields, are directional. They are described by vectors. At each point in space there is a direction to the field and a strength of the field. But not all fields are directional. For example, it's possible to plot out a field of temperature in any system of matter, such as the atmosphere or the oceans. Temperature only has an intensity, not a direction (although temperature can change with direction, that change is not a property of each point; it's only meaningful over a distance). Temperature is an example of a 'scalar' field, one which only has intensity.
The Higgs effect arises because space is filled with another scalar field - the Higgs field. The Higgs field is like an electric or magnetic charge, but it has no direction - particles can't turn in any way like the bar magnet in the magnetic field, to reduce the interaction. Certain particles - electrons, quarks, neutrinos - have an energy of interaction with the universal Higgs field and it's that energy that gives them mass. It's that mass that means they have a resistance to change in velocity - inertia.
The "Higgs Boson" itself doesn't give particles mass. It's a particle what was predicted to appear if the universal Higgs field was given a certain kind of kick. The Large Hadron Collider succeeded in giving enough kicks of the right kind to the universal Higgs field to allow the existence of the Higgs Boson to be convincingly shown.