Cryptanalysis of a feedback shift reigister with an S-box
why is this cryptosystem "keyed by the contents of the S-box" ?


It means that the content of the S-box is assumed secret, and part of the key (in addition to whatever part of the initial state is not a given).


  How to decrypt the ciphertext if we know the S-box ?


My reading is that what's shown is the keystream generator of a stream cipher, that gets combined by XOR with the plaintext outside of the drawing. We have to guess that, as in a Fibonnaci LFSR, the keystream is what's output on the right, so that the first 32 bits of the keystream are the initial state per some untold convention on bit order, likely the same as in reference material by the same author on Fibonnaci LFSR.

To decipher, the legitimate user would need to know the initial state, and repeatedly extract the right 4 bits and do three independent things with these:


XOR these with 4 bits of ciphertext to decipher 4 bits of plaintext
push these (not the ciphertext nor plaintext) on the left of the LFSR, removing them from the right
feed these in the S-box, and get 4 S-box output bits, that must be combined with XOR into the 4 bits that the above shift has moved on the right of the LFSR, and that the next step will extract.



  What does "information contained by the S-box" mean ?


That means the effective key length that the S-box represents, equivalently the entropy¹ necessary for coding an S-box (that is fully describing the input to output dunction that it implements). If there is a total of $s$ possible distinguishable S-boxes of equal probability, that's $\log_2(s)$.


  How does the invertibility of the S-box affect the bits of information it contains?


Invertibility of an S-box means that no two distinct combination of the $i$ input bits yield the same combination of the $b$ output bits. By the pigeonhole principle it implies $i\le b$ (this is the case here: we have $i=b=4$). This invertibility restricts the number of S-boxes, thus $s$ changes.



¹ If we knew how the keys are turned into an S-box (which is not the case here), we could compute the probability $p_j$ of each distinguishable S-box, and the Shannon entropy could be computed as
$$\displaystyle-\sum_{j\text{ with }p_j&gt;0}p_j\,\log_2(p_j)$$