Meteorologists say that if only they could design an accurate mathemat

It's certainly a little tricky but this is how you go about solving it:

Read the question: What would cast the most serious doubt on the meteorologists’ boast?

First of all, what is the meteorologists’ boast?
It is that if they could design an accurate model of the atmosphere, they could forecast the weather with real precision.

We need to cast doubt on this i.e. we need to say that even if they design an accurate model of the atmosphere, they still may not be able to forecast weather with real precision. Focus on this - even if they do get an accurate model, they may not forecast accurately. We don't have to worry about how difficult it is to get an accurate model, the point is that even if we do get it, it may still not be possible to forecast accurately.

Let's look at the options:

(A) The amount of energy that the Earth receives from the Sun is monitored closely and is known not to be constant.

It doesn't talk about atmosphere, model, weather or anything related to our question. We don't know how the energy from the Sun affects our variables so this option is out.

(B) Volcanic eruptions, the combustion of fossil fuels, and several other processes that also cannot be quantified with any accuracy are known to have a significant and continuing impact on the constitution of the atmosphere.
This says that there is a continuing impact from other sources on the atmosphere. So even if there is an accurate model of the atmosphere, we may not be able to predict the weather accurately because of this impact. This helps cast doubt on the meteorologists’ boast. Answer.

(C) As current models of the atmosphere are improved, even small increments in complexity will mean large increases in the number of computers required for the representation of the models.
This talks about how difficult it is to get an accurate model. That is not our concern as discussed above.

(D) Frequent and accurate data about the atmosphere collected at a large number of points both on and above the ground are a prerequisite for the construction of a good model of the atmosphere.
This talks about how difficult it is to get an accurate model. That is not our concern as discussed above.

(E) With existing models of the atmosphere, large scale weather patterns can be predicted with greater accuracy than can relatively local weather patterns.
Existing models are not our concern here. So this option is out too.