Edit: Question had been updated since my answer below. However, Kaushitaki Upanishad, being one of the 11 most important Upanishads is its own authoritative source, vs. Bhagavad Puran. Upanishads are a higher level pramANa compared to purANAs, just stating for sake of those who were not aware of this. i.e. if you have found something in an Upanishad, you have found the source and there is no reason to go look for other sources.
It is from Kaushitaki Upanishad. Verse is:
angad angat sambhavasi hrdayad abhijayase
atma vai putra-namasi sañjiva saradah satam
Meaning: Thou arisest from me, limb by limb, above all art thou born from my heart ; thou art verily my soul, my son; live thou an hundred years.
If you are asking what does it mean, it directly means what it says, i.e. the child is always born with flesh of the parents, and the Atma/Soul in the child is a split-part of the Atma/Soul of the father.
Here is the link to this upanishad.
This verse occurs on Pg 156 in this. Unfortunately, I could not find a better formated source for this upanishad online. There are of course Hindi versions for those who can read & understand Hindi:
The last word there has a sandhi (conjunction in English, but there is no equivalent for this grammatical construct in English), so that last word highlighted is in half, because the meaning I provided above ends with half of that word. Removing the sandhi "..sharadam shatamiti nAmAsya.." becomes "..sharadam shatam iti nAmAsya.." and this is the sanskrit verse provided above.
This Hindi verse is from this source, pg 57.