It's a mechanical slide adder or "addiator".
There's a somewhat detailed explanation here and a demonstration here.
For addition, for example $6 + 3,$ you insert the stylus at the $6$ position of the rightmost slot in the addition section (the upper half) and slide it down to the bottom of the slot, resulting in the configuration shown in the picture (with a $6$ showing in the rightmost circular window).
Then insert the stylus in the $3$ position and slide down again;
you'll have a $9$ in the rightmost circular window.
To add $6 + 4,$ however, you can only slide down from the $6$ position.
When you insert the stylus at the $4$ position you need to slide it all the way to the top of the slot and then, following the circular arc at the top of the slot, loop around to a notch on the left and slide it down one position, which adds $1$ to the second column.
The red dots are a clue about when you need to slide up instead of down for addition.
Subtraction of two numbers $A - B$ is similar, but after putting the digits of $A$ in the circular windows by sliding the sliders,
you select the digits of $B$ in the lower half of the device and slide up when no carrying is required or down (looping around at the bottom) when carrying is required.
The arrows in the four round windows in the picture shown appear to be an error condition; I think they are telling you that you need to perform carry operations on those columns (perhaps sliding from the zero slot all the way to the top, around and down one). Notice that in each place where there is an arrow in the window, the semicircular end of the next slot to the right doesn't have a notch at the top left for you to loop the stylus into and slide down one place. That's because the slider on the left is too far down. But it's been a very long time since I've seen one of these in real life.