I have not used the ADC1015 breakout board from Adafruit, but it looks like it uses I2C communicate to the Arduino, so you would connect that as shown in the I2C Connections or I2C “Classic” sections of the Adafruit 4-Channel ADC Breakouts guide you linked to in your first post.
How to connect the sensor to the ADC breakout board might depend on how you are using the sensor. For example, if you want to measure position, you will need to have the ability to drive the different pins high or low and take an analog reading from them at various times while taking the measurements (see the “Measuring position” section of the FSLP’s product page for more information about this). Since the ADC breakout cannot drive pins (it cannot use its input pins as outputs), you cannot simply disconnect the wire from your Arduino and connect it to the ADC breakout. Instead, you will probably want to connect D2 and SL (shown in the pictures below) to the ADC breakout board pins (e.g. A0 and A1) with a jumper wire, and leave them connected to your Arduino as well. Then when the instructions for taking the reading call for you to measure the input, you would use I2C communication to get the measurement of those pins through the ADC breakout board.
If you make a wiring diagram and post it here, I would be glad to take a look and see if I see anything obviously wrong with it. Also, if you have not tried it yet with just the Arduino, you might consider seeing if it works fine for your application; I suspect the Arduino ADC resolution (which should theoretically give you a resolution of about 0.1mm for the position measurement) is high enough for most applications with the FSLP.