Those who eat the remnants of the sacrifice, which are like nectar, go to the eternal Brahman. This world is not for the man who does not perform sacrifice; how then can he have the other, O Arjuna?
In simple words
Krishna tells Arjuna: "Those who live in the spirit of giving taste something like nectar — they move toward the eternal. But a person who never gives, who never offers anything — this world has nothing for them, let alone the next."
Word-by-word meanings
यज्ञशिष्टामृतभुजःeaters of the nectar -- the remnants of the sacrificeयान्तिgoब्रह्मBrahmanसनातनम्eternalनnotअयम्thisलोकःworldअस्तिisअयज्ञस्यof the nonsacrificerकुतःhowअन्यःotherकुरुसत्तमO best of the Kurus
4.31 यज्ञशिष्टामृतभुजः eaters of the nectar -- the remnants of the sacrifice? यान्ति go? ब्रह्म Brahman? सनातनम् eternal? न not? अयम् this? लोकः world? अस्ति is? अयज्ञस्य of the nonsacrificer? कुतः how? अन्यः other? कुरुसत्तम O best of the Kurus.Commentary They go to the eternal Brahman in course of time after attaining the knowledge of the Self through purification of the mind by performing the above sacrifices. He who does not perform any of these is not fit even for this miserable world. How then can he hope to get a better world than this (Cf.III.13)
Contemporary scholarly and practical interpretations for modern seekers.
This interpretation draws on a specific tradition and may not represent the view of any single school. For authoritative guidance within a specific tradition, seek a qualified teacher.
Modern
In ancient times, the remnants left after offering oblations to fire during sacrifice were called amrita, which the devoted would consume as a blessing from the Divine, believing that through this reverent partaking of amrita, the inner faculties could be purified.
When we contemplate the spiritual significance of this metaphor, we understand that the remaining amrita refers to the fruits obtained from the performance of the sacrifices mentioned above. The fruit of performing these sacrifices is self-mastery, or in other words, an integrated personality. Only those who have attained this fruit are fit for the practice of meditation.
A person of integrated personality alone, by attaining the equanimity of mind necessary for meditation, can recognize the infinite Brahman as one's own Self. The second line of this verse, expressed in the language of negation, further clarifies the principle stated above. Without earnest effort, there can be no development of the Self. Through inaction, no one can gain benefit in any field. Since without selfless action one does not obtain lasting fruit even in this world, what can one accomplish regarding the world beyond?
At this point, two doubts may arise in the mind. Do all these paths lead to the same goal, or do they lead to different goals? And the second doubt may be: are all these paths merely intellectual concepts of the Divine? The Blessed Lord, dispelling these doubts, speaks thus: