The paperclip challenge was originally a real story about a man who started with a paperclip and slowly but surly got things a tiny bit more rare every time he traded it until he eventually ended up owning his own house!
People have now began to incorporate it into this game as well! simply put its the act of asking another user to give them a pet of slightly higher value so they might achieve their dreamies/goals. An example of this could be an user who is doing the paperclip challenge offering a 2018 omgsc, the person who is getting the trade could cancel or possibly help them out! <3 by offering something like a 2018 very common or a 2017 omgsc. This will continue to happen until they reach their goals!
I myself do the challenge, however its often scorned upon as it requires the help of another player and may be considered as begging. However if you need anymore help in understanding the challenge or anything of the sort do not hesitate to pm me!