Did my teacher specialist about sulfur compounds made an error in part of Ramberg-Bäcklund rearragement mechanism? I have a lot of troubles with what I circled in red with the big question mark near. I am not speaking about stereochemistry but about conservation of number of hydrogen atoms, and also there is actually no reason for the electrons on the carbon near $R_1$ to attack the other carbon. In my lesson, the teacher didn't show how the extra hydrogen was removed.

Can you explain what my teacher didn't show or forgot to include? 



Thank you in advance!
As discussed above, the pathway shown on the left is incorrect because it essentially shows insertion into a $\ce{C-H}$ bond by an anion with loss of $\ce{X^-}$, which is not the easiest path for this reaction to take by a long shot.

The actual mechanism for this reaction is as shown below.


First, the non-halogenated alpha-carbon is deprotonated.
The carbanion formed arranges itself such that the it can attack the halogenated carbon and in such a way that the two R-groups are cis to each other.
Attack occurs, with loss of $\ce{X^-}$. Product is cis.
In water, due to lack of solubility, decomposition is faster than redeprotonation and the (Z)-alkene forms.
In organic solvent, base deprotonates an alpha-carbon, which can then scramble.
The main reprotonation product is the trans product.
The trans product decomposes, and the (E)-alkene is the major product.




A few notes:


The red path makes almost completely (Z) product. The blue path makes a mix of the two with the (E) product as the main product.
Purple star: the reason the anion attacks to form the cis ring is unknown, but there are several theories, apparently.
Green stars: cheletropic ring decomposition is symmetry forbidden, so this is theorized to be a two-step decomposition through either diradical bond fissions or a dipolar mechanism.