What am I looking at in Franklin's Photo 51 of DNA?
Well, you said that you...


  ...have a lot of trouble connecting the photo to the diagram.


And that's quite excusable: interpreting that X-ray image is actually very complicated . 

All the quotes and images in this answer, except the bulleted list further down, come from this paper (Lucas, 2008), which explains that historic picture in details.

Full disclosure: I always though that that image depicted the X-ray going longitudinally (that is, along the main axis) through the DNA. However, it goes transversely:


  When filamentous macromolecules are packed in a fibre along a fixed direction, the X-ray intensities diffracted by the fibre fall on the observation screen along approximately straight and equidistant lines, the so-called layer lines, perpendicular to that direction. This important concept was introduced by Michael Polanyi in 1921 for the X-ray study of cellulose. (emphasis mine)


These are the so called layer lines (I'm keeping the original paper's legend in all images):



Then, still according to the paper, Crick and others started to hypothesise how would be the X-ray diffraction in a monoatomic helix: 


  In 1952, Cochran, Crick and Vand developed an analytical theory for X-ray diffraction by a monoatomic helix. The immediate interest of their theory was to give a transparent, analytical expression of these amplitudes at a time, in 1952, when computers, if at all available, were barely capable of a brute force calculation of the total diffraction intensity.


This is the X-ray diffraction pattern in a monoatomic helix, which is quite important to understand the famous DNA diffraction image later on:



Thus, with that theoretical background, we can understand our famous image:



In this image, a X-ray diffraction image of A-DNA (left) and the more common B-DNA (right) show a periodic pattern in the layer lines.

The diffraction patterns can be interpreted as follows (source):


The layer line separation reveals the value of the polymer repeat period. The decrease of over 20% of the layer line spacing when going from A to B implies an increase of that much in the period: P = 2.8 nm for A-DNA and 3.4 nm for B-DNA.
Look at the sharp, discrete spots observed near the center of the
diffraction pattern for A-DNA along the first few layer lines, these
suggest the crystalline order in the fibre. In the high-humidity
B-DNA pattern, these crystalline spots are absent, suggesting that
the extra water molecules must have invaded the space between the DNA
molecules, freeing them from being locked into crystallites.
The thick arcs at the top and bottom of the B-DNA pattern are found at approximately 10 layer line intervals from the center, implying that B-DNA had 10 repeating units within one period of 3.4 nm. These are produced by the scattering of X-rays by the equidistant, nearly horizontal flat bases separated by 0.34 nm. The A-DNA pattern lacks these big smears, suggesting that the bases in A-DNA are not horizontal, and the number of base pairs per helical period is closer to 11.
The central cross in B-DNA represents the Saint Andrew cross expected
from a helical molecule. The large radius r (1 nm, indicated by the
meridian angle of the cross) and the absence of intensity in the
meridian diamonds indicates that the phosphate backbone is at the
periphery of the helix. This cross appears to be absent in A-DNA,
however, this is due to destructive interference from some of the
inclined base pairs.


Finally, this is an image better relating the double strand structure of the DNA (both A and B DNAs) with the X-ray diffraction image:



Reference:

Lucas, A. (2008). A-DNA and B-DNA: Comparing Their Historical X-ray Fiber Diffraction Images. Journal of Chemical Education, 85(5), p.737.