The eternal embrace which Bhagavān could never forget
Śrī Rāma accomplished the terrific task of singlehandedly eliminating the fourteen thousand demons and their chiefs Khara and Dūshaṇa, in a very short span of around one and half Muhūrta (72 minutes in modern units) in the battlefield; then seeing this all gods showered flowers over Śrī Rāma and exclaimed what a great feat! what prowess what efficiency only Lord Visnu could do this in the battle! (Śrī Vālmīki Rāmāyaṇa 3.30.32)
तं दृष्ट्वा शत्रुहन्तारं महर्षीणां सुखावहम् ।
बभूव हृष्टा वैदेही भर्तारं परिषस्वजे ॥
(श्री वाल्मीकि रामायण ३.३०.३९--४०)
taṃ dṛṣṭvā śatruhantāraṃ maharṣīṇāṃ sukhāvaham ।
babhūva hṛṣṭā vaidēhī bhartāraṃ pariṣasvajē ॥
(Śrī Vālmīki Rāmāyaṇa 3.30.39--40)
“On seeing her husband Śrī Rāma, the eliminator of enemies and accorder of the happiness to sages, Vaidēhi Śrī Sītā filled with happiness and boundless joy embraced her husband Rama, tightly with love.”
तं दृष्ट्वा शत्रुहन्तारं -- taṃ dṛṣṭvā śatruhantāraṃ (Śrī Sītā saw him, the eliminator of enemies)
Śrī Sītā seen her Lord's extraordinary prowess and scintillating performance on the battlefield when Śrī Rāma singlehandedly eliminated fourteen thousand demons with their chiefs Khara and Dūshaṇa.
These demons were so far undefeated even by Indra with all his forces. Śrī Rāma defeated such demons without any damage to his person, a + vyayam 'unscathed, undamaged... and very easily also he defeated all...' at such a Rama, Śrī Sītā saw.
(Commentators have said Kara could be killed by Śrī Rāma the supreme Brahman only, none else.)
Why Śrī Rāma eliminated them?
महर्षीणां सुखावहम् -- maharṣīṇāṃ sukhāvaham
(Out of his Sharanagat-Vatsalta,) for the happiness of the sages of Dandakaranya. Maharshis (so all other Jivas also) can now move around Daṇḍaka forest comfortably and carry out their Dharma. If softest and noblest people like sages (who harm no one) are protected from their enemies, then all Jivas are protected, it is for the happiness of all.
बभूव हृष्टा वैदेही -- babhūva hṛṣṭā vaidēhī (Śrī Sītā became filled with boundless joy)
'filled with boundless happiness', 'elated' 'exulted' 'thrilling', 'as though the inner joy is going to outburst trough every skin pores...' Such is the joy of Śrī Sītā here.
Śrī Sītā was happiest and filled with boundless joy, after seeing her Lord's extraordinary might and singlehanded victory over fourteen thousand demons, It is not just the only reason of her happiness, Śrī Sītā Jī is elated also for seeing her Lord's Sharanagat vatsalta, Her Lord is always ready to protect his devotees personally from the evildoer Rakahasas.
She being the mother of universe has happiness that her children Rishi and Maharshis have been protected by her Lord.
भर्तारं परिषस्वजे -- bhartāraṃ pariṣasvajē
This unforgettable embrace was like a victory prize gifted by the sweetest and most beautiful princess Sītā to her dearest Lord upon his terrific feat and accomplishment; when a wife takes initiative to embrace her husband with all love and happiness, husband forgets all his wounds and hardships by her affectionate embrace.
This embracing is pariṣasvajē, pari ṣa svajē 'wholly embraced.' means this embrace was not just from front, but also an embrace to her husband from back. How much she was happy!!!
She embraced him from all over and fomented his war-wounds with her warm breasts.
If not all over, she embraced her husband so tightly that Shri Rama completely came into her hands which were all around him. The most loving embrace !
Śrī Gōvindarāja comments here in his commentary on Śrī Vālmīki Rāmāyaṇa : Śrī Sītā came out of the mountain cave and embraced Śrī Rāma. This reflects that the Jiva (an individual soul) which is an amsha (shesh) on the advise of Gurudeva, after being released from worldly bondages, embraces the master, the husband Śrī Rāma (Sheshi).
Therefore, Purva-Acharyas have stressed on this verse and suggested that devotees can offer Mantra-pushpanjali (offering flowers in the end of puja as the final prayer) to Bhagavan with this mantra - Śrī Vālmīki Rāmāyaṇa 3.30.39--40.
Thus with this mantra, a devotee can feel embracing Śrī Rāma in wifely mode of devotion as like a wife embraces her husband. It is offering self to Bhagavān.
Śrī Sītā gave such a great everlasting bliss by her embrace to Śrī Rāma that Lord could not forget her embrace ever, even in his Śrī Kṛṣṇāvatāra, He remembered and was missing that unforgettable hug of Śrī Sītā even in the presence of Śrī Rādhā Jī.