In this proof that sokoban+ is pspace complete, how does the gadgets register the fact that a cell of the turing machine has changed?
In the paper they created gadget called a gate. This gadget has two entrances, from one entrance the player can open the gate, the other entrance can be traversed only if the gate its on the open state.

In the cell gadget, there are $2k + 2$ gates where $k$ is the number of states the turing machine has and the other two gates correspond to the symbols ($0$ or $1$) that the tape of the turing machine can have.

When the player enters the cell gadget, he accesses a chamber corresponding to the state the turing machine is at that moment. In the chamber, there are two gates that correspond to the two possible symbols that the turing machine can have. There, the player can open one of the gates, the opened door has to coincide with the symbol that the turing machine has in that cell.
He opens the gate by entering through one of the two entrances of the gate, he cannot access the other entrance from here. After opening the gate, he can traverse one of the two symbol gates, closing it.

After traversing the gate, the player reaches a different corridor, from that corridor he can access only the "state gates" that have the same symbol that the symbol gadget has (if they are opened), so he must have chosen to open the correct gate gadget in order to continue. Assuming that the player decided to open the correct state gadget, he can return to the same gate he opened before by entering through the second entrance of the gate (I thought that the player went to a different gate after that, the figure confused me).

After traversing the gate from this second entrance, the player closes the gate. From here he goes to a corridor that allows the player to open one of the symbols gadgets he visited previously (not a different one), and he will only be allowed to open the symbol gadget that the turing machine will write in that state and symbol configuration (if he decides not to open the symbol gate, he will be trapped if he decides to return to the gadget).