He turned off the "stasis" field
The two facehuggers were kept in stasis cylinders which were in the same Medical Lab section as Newt and Ripley.
All Burke had to do was put the stasis tubes inside the door, disengage the stasis fields and let "nature" take its course.
The novelization by Alan Dean Foster makes this a little clearer...
Gently she disengaged herself from Newt, who slept on oblivious to adult obsessions with time. Ripley made sure the small jacket was pulled up snugly around the girl's chin before turning to crawl out from beneath the cot. As she turned to roll, she caught another glimpse of the rest of the Med lab—and froze.
The row of stasis cylinders stood just inside the doorway that led toward the rest of Hadley central. Two of them were dark their tops hinged open, the stasis fields quiescent. Both were empty.
Aliens - Novelization by Alan Dean Foster
The movie isn't quite the same but we do see unbroken stasis tubes lying on the floor of the medical bay.