The Higgs effect gives particles mass, but how it gives them mass isn't explained well (so I have found after watching a talk by Leonard Susskind), and the analogies given for how the Higgs effect works are invariably wrong. The Higgs field doesn't make space seem sticky like molasses, and it doesn't work like a celebrity not being able to walk fast through a room (both common analogies). I'm not enough of a physicist to be able to go into detail, but I can explain how a field like the Higgs can give particles mass which will result in them having inertia.
Quantum fields fill space - they are everywhere even though their typical values almost everywhere may be zero. Also, fields can exist even without having definite sources. The Higgs field is present everywhere and one of its properties is that its lowest energy state - the state it is at when nothing else is going on, isn't zero. The default value of the Higgs field in our universe is not nothing. This means that the Higgs field can have an effect everywhere. Relativity says that the laws of Nature should appear the same however you are moving - whatever speed you are travelling. This means that the Higgs field appears the same for any situation of constant speed (or no speed at all).
Given that the Higgs field is everywhere and for everyone, we can then look at how a field can give something energy. To make thinks simpler, imagine that there was a magnetic field everywhere - that there was some universal North pole and an opposite South pole. Now, if you had a bar magnet in deep space, away from the magnetic fields of planets, stars and galaxies, it would try and line up with the universal magnetic field, with the magnet's North pole pointing to the universal South and the magnet's South pole pointing to the universal North. This would be the lowest energy state of the magnet. Now, if you turned the magnet 180 degrees so its North pointed to the universal North and its South pointed to the universal South, the magnet would be at a higher energy (it takes energy to turn it to that direction). Now, as Einstein said E = mc2 : energy has mass. The magnet in this new position would be more massive, and so it would take more force to accelerate it. The effect would be extremely small, but it would be there.
Now, the Higgs field isn't like a magnetic field. For one thing, it has no direction (it's a 'scalar' field). But, it can interact with things in a similar way so as to give them more energy than they would have than if the Higgs field were not there, and that energy means that such things have mass that they would not have. It's nothing like the analogies listed above - the Higgs field doesn't slow anything down, or get in the way, it simply adds mass-energy to particles because they interact with the Higgs field.
There is still much to be discovered - different particles interact with the Higgs field to difference extends, and so there is a (vast) range of masses of the known fundamental particles. But, why particles gain mass from the Higgs field isn't particularly hard to understand.