I think this question, with editing, has a lot to do with mathematics education, in that it points to the problem of learning to read and interpret mathematical writing. I would instead ask something like: "How does one glean the ideas in a mathematical paper without reading it line by line?"
Now for a somewhat rambling answer to this question:
Note that mathematical work is steeped in context, and that without familiarity with the standard ideas and tricks surrounding the paper in question, it is nearly impossible to find the idea in the paper.
Reading line by line can allow the reader to perhaps, with a LOT of work, reconstruct a consistent mental model that reproduces the results of the paper (perhaps this is what mathematics is, making such models). What is meant by "first finding the idea" is that much of what is written in a paper is standard in the community of persons working on the problems in question. The new idea in the paper stands out against the standard "background noise". In order to find it, you have to first know what is standard.
Stated another way: If you are using only standard techniques tried by many experts to solve a reticent problem, then you are very likely to be wrong. So your proposed solution is first evaluated by whether it is novel against the "background noise" and whether it hangs together logically as a strategy. A mathematical proof has a basic logical outline that contains "the idea", and much of the detail needed to complete the idea is necessary tinkering with standard techniques to make it all work out. I hope this helps.
As for mathematics education, this is very important insofar as it encodes "big picture thinking", as opposed to "algorithmic nose-following" that many think is mathematics. See Paul Zeitz's book "The Art and Craft of Problem Solving" for a discussion of strategy, tactic and tool.
Without a sense of the living ideas encrypted in the writing of a mathematical paper, which sadly are not found in traditional mathematical writing, one is flying blind. The belief that reading papers line by line will yield an understanding of a subject is pernicious, and this question (if properly asked) has the educational benefit of dealing with that pernicious belief.
(Sorry for the behemoth comments…which now appear as the present answer.)