You have some sort of plastic water line that works with a very specific type of stop valve that looks like the the one in the back of the last picture. The plastic seats on the 1/4" valve body and the compression nut keeps it tight.
For some reason, it looks like you don't have the other side of that valve, and you are trying to connect it to a 1/2 x 3/8 valve with a 3/8 x 1/4 reducer.
If I had to do this myself, I'd either replace the stop valve with a 1/2 x 1/4, and see if you could get it to work with no brass ferrule, or I'd cut the water line straight behind the compression nut, use a double end press-fit connector such as this one: https://www.homedepot.com/p/John-Guest-1-4-in-O-D-x-1-4-in-O-D-NPTF-Polypropylene-Push-to-Connect-Coupling-Fitting-803149/300753474 combined with a short piece of copper 1/4 tubing to your existing compression fitting.
Alternatively, they make metal inserts that stabilize the plastic water line, and you can just use the compression fitting that you have on that valve like you were trying to.
Edit: Technique for using plastic compression fittings with brass reinforcement inserts is here: https://www.youtube.com/watch?time_continue=127&v=aFpZfr0YQ28&feature=emb_logo