Although I hear they were caught cheating off of Rosalind, who sat at a desk in front of them.
The 'Cool S' is a stylized drawing of the letter "S". It is a popular doodle among teenagers and pre-teens as it can be quickly hashed out using six vertical lines which are then connected with an appropriate pattern of diagonal lines. The drawing has been around for a very long time, and may be independently discovered simply because it is a logical progression of combining a series of straight lines in an interesting manner.
Randall first draws the steps to make the Cool S. Then he draws a chain of repeats of this pattern, excluding the end caps, making something that looks like a twisted rope. Finally he separates the two strands and adds cross bars, creating something that turns out to have a visual similarity to the discovered helical structure of DNA. However, although the final diagram is a double helix, the chirality, or "handedness", is backwards: an actual DNA molecule usually winds in the opposite direction (clockwise, if you're looking at it end-on). (The tutorial at How to (correctly) draw DNA describes this in more detail.) It is possible for left-handed DNA segments, called Z-DNA, to form in purine-pyrimidine alternating regions; but this is much less common than the more typical conformations of B-DNA and A-DNA. (B-DNA is the most common and is elongated with accentuated major grooves: see Nucleic acid structure. This asymmetry can be seen in the spacing between strand segments, which alternates between short and long distances, as opposed to A-DNA and Z-DNA where these distances are much more uniform.)
He posits that the helical shape of DNA was originally discovered when somebody decided to doodle this extended S pattern.
Understanding the shape of the DNA molecule was an important step towards understanding how it duplicates itself and serves as a template for RNA. In real life, Francis Crick and James Watson were awarded a Nobel Prize for this discovery.
The title text is a reference to Rosalind Franklin, who made a material contribution to the discovery of DNA but was controversially not included in the Nobel Prize. Many are quick to assume she was excluded simply due to sexism, although she had also been dead for five years (Nobel rules now prohibit posthumous nominations (though this statute was not formally in effect until 1974)). Franklin's boss at the time of the discovery, Maurice Wilkins, was also named on the prize, and Nobel rules also forbid splitting of Prizes more than three ways. Wilkins had shared some of Franklin's data with Watson, who then shared what he saw with Crick. None of the three men ever told Franklin that Watson and Crick had based their model on her data. Fortunately, an international astrobiology programme, ExoMars, has since named a Mars rover after her!
This account is reminiscent of how August Kekulé said he identified the ring structure of the benzene molecule. He claimed to have had a dream in which the atoms were moving around, with the last grabbing onto the first like the classic image of a snake grabbing its tail, forming a ring of six lines.