When. Said to the parents on the birth of a child, as a congratulation. Its source here is an-Nawawi's al-Adhkar, where it is given as the wording to greet a new parent with rather than as a transmitted prophetic text — worth knowing, and the page says so rather than implying otherwise.
The words. Four clauses, and they are not four ways of saying the same thing.
Bāraka Allāhu laka fī al-mawhūbi lak — "in what has been given to you". Mawhūb is from the same root as the gift in Ibrahim's supplication: the child is named as something bestowed, not achieved.
Wa shakarta al-wāhib — "and may you thank the Giver". The second clause turns from the gift to the One who gave it, which is the move the whole supplication is built around.
Wa balagha ashuddah — "and may he reach his full strength". Ashudd is the Qur'anic word for the age at which a person comes into their own.
Wa ruziqta birrah — "and may you be provided with his devotion". The last clause asks for something the parents receive back, and it asks for it as rizq — provision, not entitlement.
What is being asked. Read together the four move outward and back: blessing on the gift, gratitude to the Giver, the child's own maturity, and finally what returns to the parents. The order is the point.