Is there a way to split a double wall plug into two separate ones?
You may notice metal tabs between the top and bottom screw on each side of the outlet. These short the top and bottom outlets together. One set of black/white wires provide power to both outlets while the other set of wires continues on to other outlets. This why, when you disconnected the second set of wires, other outlets stopped working. The original installer used the outlet itself as a jumper between the sets of wires.

In cases where the two outlets are powered separately (usually when one is controlled by a switch), these tabs are broken off to isolate the two outlets.

If you want to split the feed to a new outlet, one way would be to connect the new wires to the screws on the outlet. This way, the outlet acts as a jumper between all three sets.

However, I'd note that while the "back-stab" connections to the outlet are legal, many electricians here do not use them and do not trust them. The better way would be to remove the wires from the back-stabs and connect the two black wires, together with the new black wire and a short length of wire (a pig-tail) using a properly sized wire-nut. Then connect the pig-tail to the outlet screw. Do the same for the white wires.

Finally, I need to point out that working with mains voltage is very dangerous. You've already admitted that you have limited knowledge of the field. The voltage you're working with can easily kill you. Also, not connecting the wires well could cause a bad connection that could later heat up and cause a fire! I highly recommend you find an electrician or so least and experienced handyman to do the work!