What is a best fit branching fraction? While describing the decay of Higgs to final states which are not allowed by standard model many papers report numbers like "The Branching fraction $B(H \rightarrow XY)<1\%$ at $95\%$ confidence level (CL), while the best fit branching ratio is $B(H \rightarrow XY)=(0.8_{-0.2}^{+0.2})\%$".
What is a best fit branching ratio and what do the numbers mean?
 A: They are reporting the same results two ways. 
One way is to say "we're quite sure [the thing we're measuring] is less than 1%", and the other way is saying "our best guess [from fitting a theoretical expectation to the data is] at the value of [the thing we're measuring] is 0.8% [with an error range running from 0.6% to 1.0%]".
The former is almost certainly a model independent statistical counting statement, while the latter depends on a particular theoretical model that used with the data to find optimal values of some set of interesting parameters.
The author of that passage expected you to notice that the upper bound from the pure counting and the upper bound from the fit are the same—a fact which gives the results a extra bit of 'well, that makes sense'-itude.

Now, "branching fraction". When a reaction or decay can go to more than one possible result, the question of how often does it go to each result is interesting, and the answer is usually quoted in terms of branching fraction
$$f_{\text{outcome }i} = \frac{\text{number of events with outcome }i}{\text{total number of event}} \;.$$
