When. Said for one's children, and said by people who do not yet have any. It is Qur'an, Ibrahim 14:40, spoken by Ibrahim in old age.
The words. Muqimas-salah is not "one who prays". Iqamah is from the root q-w-m, to stand, in a form meaning to make something stand upright and keep it there. Arabic never says "do the prayer" in the Qur'an; it says establish it, as one establishes a building or an institution. The request is for prayer to be something erected and maintained, not an act performed.
Then wa min dhurriyyati — "and from my offspring". Grammatically this is incomplete on purpose: and from my descendants, likewise. And note min, from among — some of them. Even here the request is measured; it does not claim every descendant.
Who is speaking. Ibrahim asks this as an old man with a son, in a passage about a valley with no crops. He asks for prayer to be established in his line before asking for anything else, and the request outlives him by every generation that has said it since.
The closing line. Rabbana wa taqabbal du'a — accept my supplication — is a request about the request. It appears after the asking is done, and it concedes that being answered is not automatic.
Why it is said for children. A parent can compel attendance and cannot compel establishment. The verse asks for the part that is beyond a parent's reach, which is why it is a supplication rather than a plan.