Here is my attempt:
"From food all beings are created"
We learned from scriptures that when people die, the spirit soul falls back to the earth via rain. Rain falls onto crop fields and the spirit soul enters the crops, grains, etc. These crops and grains are consumed by people and the spirit soul enters a male's semen through food. During intercourse, the spirit soul is carried to the womb via the semen and that is how we are born. Hence the consumption of crops and grains by male organism results in the rebirth of the spirit soul
"From rain food is created"
The meaning is clear because, since sufficient rainfall leads to crop growth and this leads to food supply.
"Yagna gives birth to rain"
When we do yagna properly we satisfy the demigods, who reward us. Yagnas that satisfy Indra led to ample rainfall.
"Rain is born out of action"
Carrying out an yagna properly is an example of action
Hence the direct meaning of this quote is "action ==> yagna ==> Indra's satisfaction ==> rainfall (which carried water and spirit souls) ==> crops grow and spirit souls reside there ==> crops are consumed and spirit soul enters male's semen ==> male and female union leads to the rebirth of the spirit soul.
The translation looks correct to me.
Here is a purport of this Gita verse in a purport in The Bhagavat Purana, Canto 7 chapter 12 of Prabhupada's version of this Purana:
annād bhavanti bhūtāni
yajñād bhavati parjanyo
"All living bodies subsist on food grains, which are produced from rains.
Rains are produced by performance of yajña [sacrifice], and yajña is born
of prescribed duties." (Bg. 3.14)
When food grains are sufficiently
produced, both animals and human beings can be nourished without
difficulty for their maintenance. This is nature's arrangement. prakṛteḥ
kriyamāṇa-ni guṇaiḥ karmāṇi sarvaśaḥ. Everyone is acting under the
influence of material nature, and only fools think they can improve upon
what God has created. The householders are specifically responsible for
seeing that the laws of the Supreme Personality of Godhead are maintained,
without fighting between men, communities, societies or nations. Human
society should properly utilize the gifts of God, especially the food grains
that grow because of rain falling from the sky. As stated in Bhagavad-gītā,
yajñād bhavati parjanyaḥ [Bg. 3.14]. So that rainfall will be regulated,
humanity should perform yajñas, sacrifices. Yajñas were previously
performed with offerings of oblations of ghee and food grains, but in this
age, of course, this is no longer possible, for the production of ghee and
food grains has diminished because of the sinful life of human society