There are several scenarios here depending on what you actually want to do with your assets:
1. If you want to delete your asset from the entire Pics.io library, you just need to hit the trash bin icon on the left side-panel, in the Infopanel or on the thumbnail of the asset. Read more about deleting files from Pics.io here.
2. If you simply want to remove an asset from one particular collection, but keep it in all other places in Pics.io library, you need to detach this file from this collection. To do so, just open your asset and find Collections section in the Infopanel. Choose a collection you want to detach and click on the cross next to its name. Alternatively, you can do the same on the thumbnail of your file.
In this way you will remove this asset from this particular collection, but save it in all other collections or lightboards it may belong to.
NB If the detached asset belonged to only one collection, it is moved to root folder. You can find it later with the help of the button "Don’t show assets in nested collections".