Before I do that, however, let me give an entry point to those of you who do not even know what a branching fraction is. Subatomic particles are all subject to disintegration to lighter bodies, when not prevented from doing so by unbreakable conservation rules (such as energy conservation, or charge conservation). A particle which has more choice than others in how to decay is the Higgs boson: since it "couples" to all massive particles, it can decay into all of them, provided the total mass of the daughters does not exceed its own. A "branching fraction", or "branching ratio", then is a number with value between 0 and 1, which describes how probable is the decay into one particular final state. If there are many possibilities, then the sum of all branching fractions must make 1.0, which just says that the decay must occur into one of those final states. Now, let me paste the comment and the answer below.
Comment (by bozox):
In my unenlightened opinion, it has to do mostly with the rate of the Higgs mass to the immediate decay target(s) mass. If the Higgs boson is lighter than the decay target, the likeness of that channel of decay falls exponentially with mass difference. If the Higgs mass is greater, the probability still falls (the half-rememberd term "resonance" comes to my mind), although I'd think that the rate of likeness falls slower than exponentially.Answer:
Now, other than masses, the decay probability is affected by the value of the coupling constant. Also, for quark/gluon decay channels, the likeness is enhanced by the fact that they are colored, and each colored variety counts as a separate decay channel for probability calculation purposes.
Once you sum up all the possible probabilities for a given Higgs mass, you can calculate the branching rate of different channels.
And this is the sum total of my particle physics knowledge. That's what reading Tommaso's blog on a regular basis does to a person.
You hit on the two key concepts behind the shape of the branching fraction curves, that of phase space and that of coupling strength, but I need to put some order there.
First, phase space, which is a measure of how the final state particles can share the excess energy from the decay (mass of decaying particle minus mass of products), energy which appears as kinetic (or as we say, momentum). If a body has rest mass M, the sum of the masses of the decay products is strictly smaller than M; equality would mean "zero phase space", i.e. that all products remain still with no momentum in the parent particles' rest frame: this is not possible, since the reaction rate is zero there. Take the opposite case: a decay of M into two bodies of mass m<<M. Here the phase space is very large, and the reaction occurs freely. Intermediate cases affect the decay rate with a certain power of the phase space factor (the energy that can be imparted to the decay products).
So if the Higgs must decay to two W or two Z bosons, and its mass is 190 GeV, the WW decay is favoured wrt the ZZ one because of the larger phase space available.
Other factors are however playing in with much stronger impact. You mention the colour of the quarks as one factor to keep in mind, and you are right: if the Higgs can decay to tau lepton pairs and charm quark pairs, the two reactions have a similar phase space (the tau weighs 1.77 GeV, the charm 1.2 GeV, both almost irrelevant with respect to a Higgs mass of say 120 GeV), but the charm quark comes in three kinds, so it receives a factor of three in relative rate. However, the coupling goes with the square of the particle mass, favouring the tau decay.
Further, one must factor in a subtlety: what matters for the Higgs decay to charm pairs is not the charm mass at zero process energy -estimated as 1.2 GeV- but rather, the mass "evolved" to the energy of the reaction, i.e. the Higgs boson mass. So the effective mass of the charm quark "seen" by the decaying Higgs is much smaller (I forgot the actual number, but well less than 1 GeV). From this, it follows that the tau lepton decay rate is higher than the charm decay rate despite the factor of three from colour. You can check the two branching fraction below:
As for the H->bb decay, it of course is much higher because of the larger mass of the bottom quark. However, note how the decays to WW takes over once the two particles have sufficient phase space. Note that this decay is not zero for Higgs mass below 2M_W: this is because one of the two W bosons can be produced "off mass shell", and this still has a relevant probability of happening, given the very large coupling of the Higgs to W bosons.
A further feature is the fact that the ratio between the decay rate to WW and ZZ is larger than one everywhere, and it tends to 3:1. The reason for this is in the explicit coupling terms of the Higgs to W and Z bosons in the electroweak Lagrangian.
Much more would need to be said about the figure, but space and time are tyrants... Stuff for another post.