The stitch used for the bird motifs is called linen stitch, because it emulates the binding most commonly found in linen cloth: tabby. It is simple darning, first filling in threads in one direction. Then, in a 90° angle to that, you weave the thread in with your needle, alternately going over and under the threads already in (including the threads of the meshwork).
In case you have never tried darning in this way before, you can have a look at Thérèse de Dillmont's instructions, straight out of her wonderful book "Encyclopedia of Needlework". And if you don't know her book yet, consider getting a copy - either an old one, or one of the reprints. Or go for the online version at encyclopediaofneedlework.com. In any case, don't miss out on this book: If I could choose only a single book for textile crafts to keep, it would be this one.
Bonus pic: The netting needle tip, without thread on.