Why do we need the quantum eraser experiment? I have read many articles about the quantum eraser experiment and I am still trying to figure out why we need it. 
In this sketch I show the experiment with and without the second beam splitter.
What I can't figure out is why we wouldn't expect interference from the setup on the right which includes a 2nd beam splitter. In that setup each detector is receiving photons from two separate sources. Detector 1 receives photons reflected from the 2nd splitter and photons reflected from a farther radius point at mirror B. Detector 2 receives photons reflected from the 2nd beam splitter and photons reflecting from from a farther radius point at mirror C. There should be interference at either detector. In the first setup (the one on the left), each detector only receives photons from one source so there should not be interference. What am I missing?
 A: The setup you have drawn shows no 'eraser' part of the quantum eraser experiment. One way to add an eraser is using polarizing elements in the Mach-Zehnder setup you have drawn on the right.
Try reading this article :
http://www2.optics.rochester.edu/workgroups/lukishova/QuantumOpticsLab/homepage/snyderlapuma.pdf
To answer why we need the quantum eraser experiment, it is more a proof of principle to show that an interference pattern is destroyed if we have 'which-path' information. (This is Neils Bohr's complementarity principle which states one cannot examine both particle and wave properties simultaneously. Here, the 'particle' property = 'which path' information and 'wave' property = interference). You can find a detailed description in the link above.
A: We need quantum eraser experiments as experimental evidence for the complementarity of interference and which-path information for quantum objects.
For classical waves (sound, light at sufficient intensities), we get interference whenever two waves overlap. This is what happens in the experiment on the right of your sketch. On the left, there can't be any interference because the waves don't overlap at the location where your detector is.
The following experiment is done with a laser. The important thing is that it works exactly the same with a single photon source or for quantum objects with mass, for example electrons.

The red lines show the light path, the setup is the same as in your right sketch, the screen/detector is at the top.
The parts circled in green are polarizers.


*

*Without any polarizers you get interference because the light travels both paths and they are combined again by the second beam splitter. The same happens for single photons. They "must have taken both ways" to produce interference, but this statement has to be put in huge quotation marks.

*When you put in the two polarizers at the bottom and set their polarization angles perpendicular to each other, the interference disappears. This can easily explained for light as electromagnetical wave because perpendicular fields can't interfere. The first funny part is that the same happens if you send single photons through the interferometer. This can be explained by the fact that you can have either interference or which-path information. With the two polarizers in place, you can tell at the detector which path a photon has taken, but you can't get interference. If the angle of polarization is NE-SW, the photon took the upper path, for NW-SE-polarization, the photon must come from the bottom path.

*It gets even funnier when you also put the third polarizer in place. That is the quantum eraser. Before the polarizer, you have which-path information and you can tell which path the photon has taken. After the polarizer, all photons are polarized along N-S, so you can no longer tell which path was taken. And you also get interference.

