How to judge the extent to which experiment supports a theory? I am aware of the 'one sigma', 'two sigma' etc 'level of significance'. However this description of the significance of a result with sigmas seems to say 'If this other theory was not correct, then there would be only an $x\%$ chance of this experimental result'. For example, with the Higgs boson 'discovery', the result was significant to the 5 sigma level because there was only a one in 3.5 million chance of the Higgs boson-like 'hump' that was seen simply being part of the background noise.
Suppose you are trying to test how closely a value predicted by a theory matches experiment. You perform the experiment and get, say, 0.998$\pm$0.014 and the predicted value is 1. Is there anything that you can say about this particular data to judge how closely the experiment fits with the data (e.g. can you divide the error up and say that the experimental value is within a quarter of a standard error)? Or are you simply limited with this given data to saying that the experiment is consistent with the theoretical prediction within the error, and then to judge more closely how well the prediction matches with the real world you would need to conduct higher precision experiments or take more repeats to reduce the standard error in your experimental value?
 A: As you've noticed, traditional physics analyses, including the discovery of the Higgs boson, use frequentist hypothesis testing. One calculates the probability of obtaining data at least as 'extreme' as that observed, were the null hypothesis (e.g., the standard model without a Higgs boson) true. This is known as a $p$-value. If the $p$-value is sufficiently small, we decide to reject the null hypothesis. 
This says nothing directly about the plausibility of the alternative or null hypotheses, and, even if the null hypothesis is true, the $p$-value won't indicate that it's favoured by data. Even asymptomatically, it wouldn't converge, but make a random walk between zero and one.
There is, however, an alternative framework - Bayesian statistics - in which one may directly calculate the change in relative plausibility of models in light of data. This requires at least two competing explanations of data. 
Suppose you measured a quantity, $D =0.998±0.014$. With Bayesian statistics,  you can find the change in relative plausibility of two competing models, in light of that measurement. See e.g. my article about the notorious diphoton anomaly for a pedagogical example. The plausibility of a new massive resonance relative to the standard model increased by about 10 in light of early run-2 LHC data.
