OED has

> **3 a.** To disburden (a woman) of the fœtus, to bring to childbirth; in *pass.,* to give birth to a child or offspring. Rarely said of beasts. (The active is late and chiefly in obstetrical use.)

which is a use of *deliver* in an older [and obsolete] sense meaning *relieve*:

> †**2. a.** To free, rid, divest, clear  *(a) of,  (b) from.*

The language is formal, and perhaps a little archaic. A bit like monarchy, really.