It means "past two years of age".
Back in days before they had the same level of medical expertise that we have now, it was frequently the case that children died of the common childhood diseases which (in general, in the Western world) we no longer have to bear, because of the universal programes of vaccination that have been set up.
What is being stated in this article is that researchers believe they may have worked out what Anne's medical problems were that caused her children to be even less able to survive babyhood than usual even for that day and age.
(As has been pointed out in a comment to this answer, "Anne" is of course "Queen Anne" who was reigning monarch of England and Wales at the end of the 17th century, and whose demise signalled the end of the Stuart dynasty and the merging of England and Wales with the hitherto independent Scotland into one nation.)