Your enquiry is about the story of this incidence. Here in the above verse we find the description of Indra taking away one wheel of the Sun's chariot for his friend Kutsa to defeat his enemies. . But there is also a background to this story , which is associated with Rishi Etesha एतेश and the the reason can be found in various mantras of vedas ,which tells us the story in brief.
There is some story of rishi Etesha mentioned in Rig-Veda as well as in Atharva-Veda , which tells us that rishi Etesha was competing /fighting with Surya in a chariot race as described in Atharva-Veda - Kanda 20 -Sukta 35- Mantra 15.
एवा ते हरियोजना सुवृक्त्रिन्द्र ब्रह्माणि गोतमासो अक्रन |
विश्वपेशसं धियं धा: प्रातर्मक्षु धियावसुर्जगम्यात् || AV 20.35.15
Now unto him of these things hath been given what he, who rules alone
o'er much, electeth. Indra helped Etasa, the Soma presser,
contending in the chariot-race with Sūrya.
And here is Hindi Translation of above verse.
Similar story also can be found in Rig-Veda Mandala 5-Sukta 31- Mantra 11 .
सूरच्श्री परितक्म्यायां पूर्वं करदुपरं जूजुवांसम् |
सं रिणाति पुरो दधत्सनिष्यति क्रतुं न : || RV. 5.31.11||
sūraś cid ratham paritakmyāyām pūrvaṃ karad uparaṃ jūjuvāṃsam |
bharac cakram etaśaḥ saṃ riṇāti puro dadhat saniṣyati kratuṃ naḥ ||
11 When night was near its close he carried forward e’en the Sun's
chariot backward in its running. Etaga brought his wheel and firmly
stays it: setting it eastward he shall give us courage.
When the Indra and Rishi Etesha were competing Indra did stopped the Surya's chariot which was coming from the front of the Rishi Etesha to protect him as described in Rig-Veda Mandala 5 - Sukata 29-Mantra 5.
अध क्रत्वा मघवन्तुभ्यं देवा अनु विश्वे अददुः सोमपेयम् |
सूर्यस्य हरितः पतन्तीः पुरः सतीर उपरा एतशे कः ||
adha kratvā maghavan tubhyaṃ devā anu viśve adaduḥ somapeyam |
yat sūryasya haritaḥ patantīḥ puraḥ satīr uparā etaśe kaḥ ||
5 Thus all the Gods, O Maghavan, delivered to thee of their free will
the draught of Soma; When thou for Etaśa didst cause to tarry the
flying mares of Sūrya racing forward.
This is clear from another translation of Rig-Veda By Dr. Ganga Sahai Sharma - See Page 622.
In all this Indra did tried to protect Rishi Etesha from Surya's chariot. Rig-Veda Mandala 4- Sukta 30 - Mantra -6
tells us that
यत्रोत मर्त्याय कमरिणा इन्द्र सूर्यम | प्राव: शचीभिरेतशम् || RV
yatrota bādhitebhyaś cakraṃ kutsāya yudhyate | muṣāya indra sūryam ||
6 When also for a mortal man, Indra, thou speddest forth the Sun, And
holpest Etaśa with might.
Here is Hindi translation of above.
And to protect Rishi Indra did stopped the chariot of Surya and took away one wheel of the same . So from above quoted mantra Rig-Veda 5.31.11
. We come to know that Indra did took away the wheel in order to protect Rishi Etesha. - See Page 628
Brief account of the story
- There was certain vedic Rishi named Etesha. Long time ago he was competing/fighting with surya. Surya was riding in his chariot at that time. When Surya's chariot was coming from the front of the Rishi , Indra stopped the chariot and pulled one wheel of his chariot to protect Rishi Etesha. And later with this wheel did slayed his foes. So Indra did took away the wheel to protect Rishi Etesha as well as with it to slay his enemies.