Dinesh, this subject is also that I do not understand in detail and would like to understand better (especially the LHY correction), but I can say this much: BEC with attractive interaction are unstable in general. Atoms will rush to clump on top of another and decay via three-body loss. However, if there is another, repulsive interaction, which is effected by quantum fluctuation (to which the LHY correction is associated) , it will cancel out the attractive interaction at some length scale. Hence the BEC stabilizes into small droplets as per the paper you cite.
Note that the dipolar interaction (which is studied in your reference) is not crucial. If I recall correctly, its role is to provide enough repulsive interaction to reduce the large attractive s-wave interaction, so that the net interaction (without quantum fluctuation) is a small attractive interaction. LHY correction is quite tiny, so you wouldn't observe its effect unless the attractive term you are trying to balance is already small.
Here is an example of stable BEC droplet without dipolar interaction, by researchers from ICFO: https://arxiv.org/abs/1708.07806