Poynts are places where an item is laced to another item, eg armour is laced together with armour poynts. It was generally the method used to hold certain types of clothes on eg trews to a doublet and sleeves to the bodice of a gown.
Why is it spelled poynts?
In period it was speled in the usual multitude of ways. These days either points or poynts is equally valid, but if you use poynts - it's easier to distinguish it from any other meaning of the word point.