Poynts are places where an item is laced to another item, e.g. armour is laced together with armour poynts. It was generally the method used to hold certain types of clothes on e.g. hose to a doublet and sleeves to the bodice of a gown.
Poynts can also refer to the lacing itself - which was generally made of some form of braid. This braid had to be narrow enough to pass through the holes created in the garment/armour, and also had to be strong and hard-wearing enough to serve the purpose for which it was created (especially the case with armour poynts).
Poynts also needed aglets both to stop the end of the braid from fraying, and also to assist in lacing the poynt through the holes.
Why is it spelt poynts?
In period it was spelled 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.