Short answer: I believe Paul Cohen's piece on his own development of forcing provides an answer to your question. I will elaborate further below the break, but you can find his article written up as:
Cohen, P. (2002). The discovery of forcing. Rocky Mountain Journal of Mathematics, 32(4).
Elaboration: A brief idea of the "big picture" seen by Cohen (here I assume, perhaps incorrectly, that you mean "big idea" rather than a literal drawing) is as follows:
$1.$ Believe that philosophical arguments can be applied to questions in Number Theory;
$2.$ Believe there are many different models of mathematics (cf. the Löwenheim–Skolem theorem); and
$3.$ Believe in the existence of some sort of "decision procedure" that simplifies complex statements until they are decidable.
These "big picture" ideas are what evolved and interacted with one another in leading Cohen to his proof of the independence of the Continuum Hypothesis. It might be worth remarking that another important belief, more related to affect but also mathematically relevant, held by Cohen was that:
$4.$ The Continuum Hypothesis was the sort of problem Cohen would like to tackle, and he was determined not to lose the chance to resolve such an outstanding question.
Here are some quotations by Cohen to support items $1$ to $4$, respectively:
In set theory when dealing with fundamental
questions, one often has a kind of philosophical
basis or conviction, rooted in intuition, which will
suggest the technical development of theorems.
Basically [the Continuum Hypothesis] was not really an enormously
involved combinatorial problem; it was a philosophical idea.
We will never speak about proofs but only about models.
and, expatiating further,
The existence of many possible models of
mathematics is difficult to accept upon first
encounter, so that a possible reaction may very
well be that somehow axiomatic set theory does not
correspond to an intuitive picture of the mathematical
universe, and that these results are not really part of
normal mathematics . . . . I can assure you that, even
in my own work, one of the most difficult parts of
proving independence results was to overcome the
psychological fear of thinking about the existence of
various models of set theory as being natural objects
in mathematics about which one could use natural
On the relation of $1$ and $2$, Cohen remarks:
Of course, in the final form, it is very difficult to
separate what is theoretic and what is syntactical. As
I struggled to make these ideas precise, I vacillated
between two approaches: the model theoretic,
which I regarded as roughly more mathematical,
and the syntactical-forcing, which I thought as more
If you actually wrote down the
rules of deduction—why couldn’t you in principle
get a decision procedure? I had in mind a kind of
procedure which would gradually reduce statements
to simpler and simpler statements.
Because of my interest in number theory, however,
I did become spontaneously interested in the
idea of finding a decision procedure for certain
identities... I saw that the first problem would
be to develop some kind of formal system and then
make an inductive analysis of the complexity of
statements. In a remarkable twist this crude idea was
to resurface in the method of ‘forcing’ that I invented
in my proof of the independence of the continuum
$4.$ This concerns, in particular, Friedberg's solution to Post's Problem using the priority method, and parallels between this and Cohen's work on ~CH using forcing. See the earlier MO question here, and, e.g., Cohen's remark:
It was exactly the kind of thing I would like to have done. I mentally resolved that I would not let an opportunity like that pass me again.
You can find the points made above in a slightly more (or less?) organized write-up here.
As a final, general remark: Having a big picture in mind when proving something occurs, of course, outside of mathematics as well. The best exposition of this (of which I am aware) is Howard Gruber's (1974) book Darwin on Man, which explores how Charles Darwin developed his theory of evolution. Though Origin of Species is structured as a presentation of overwhelming evidence that simply led Darwin to his final theory, the reality is that he was constantly theorizing along the way. Gruber explores this by going through previously unread notebooks of Darwin's, and includes a discussion of a literal picture that turns up repeatedly - in different forms - of a branching tree. Although this books concerns work outside of mathematics, I think reading it can be both informative about "creative thinking" in general, and can also provide a way for those who are mathematically-inclined to try and write up the biography of a mathematical theorem and the big ideas/big picture behind it.