There are several things the buddy does and some things your RO/DI system does now that will be replaced by the buddy.
Think of the system as 3 parts. One is the filters before the membrane, two is the membrane, and 3 is the DI after the membrane. The filters flow all the water through and hopefully trap some of the stuff in the water. Then the water goes to the membrane which has 3 holes, input, clean output, and waste. Pressure forces some water through the membrane to the clean output, most of the stuff in water is bigger than water and will not go through the membrane. That stuff, along with some water comes out the waste. Your RO/DI system has a flow restrictor on the waste side that increases the pressure and makes more water end up on the clean output. The waste water ends up in the drain and the clean water goes on to the DI to get even more stuff taken out. There is also a valve that when the output is stopped the input is also stopped.
To install the buddy you will need to remove the flow restrictor and the auto shutoff valve, the buddy will handle these things. The hose that goes from the 3rd filter to the auto shutoff you will send to From Pre-Filters on the buddy. You will have a hose from Supply to membrane on the buddy to the input of the membrane. The membrane waste will go to From Membrane Waste on the buddy. Out to Drain on the buddy will go to the drain. The output of the membrane currently goes to the auto shut off make it go to Filtered water on the buddy. Out to float valve on the buddy goes to where the other tube from the auto shutoff went, to the DI or the float valve .
Once the buddy is installed it will take the output from the prefilters and pump it up and send it to the input of the membrane. It takes the waste output from the membrane and restricts it to keep the pressure high befor sending it on to the drain. It takes the clean output from the membrane and sends it on. When it senses that the clean output is stopped it turns off its pump and the input and waits until you want more water to start up again. When it first realizes you want water, because the output opens up, it runs a little bit without the flow restrictor on the waste line from the membrane. This allows more water to flow through and clean some of the trash off the membrane to, hopefully, make it last longer.
