The black thread is a symbolic formulation of the darkness of the night (light) and the white thread is a symbolic formulation of the brightness of the day (fajr).
(To be found for example in Mokhtassar Tafsir ibn Kathir)
So once you can make a difference between darkness and brightness you should start your fast.
I found some Ahadith in Sahih Muslim and Sahih al-Bukhary and Sunan abi Dawod which quote this, espacially the Versions narrated by Muslim and abi Dawod support the tafsir above. Here's the one from Sahih Muslim:
'Adi b. Hatim (Allah be pleased with him) reported that when (this
verse) was revealed: " Until the white streak of the dawn becomes
distinct from the dark streak" (2: 187) Adi b. Hatim said: Messenger
of Allah, verily I keep underneath my pillow two strings, one white
and the other black, by which I distinguish night from dawn. Upon this
the Messenger of Allah (ﷺ) said: Your pillow seems to be very large.
For the word khait implies the blackness of the night and the
whiteness of the dawn.
حَدَّثَنَا أَبُو بَكْرِ بْنُ أَبِي شَيْبَةَ، حَدَّثَنَا عَبْدُ اللَّهِ
بْنُ إِدْرِيسَ، عَنْ حُصَيْنٍ، عَنِ الشَّعْبِيِّ، عَنْ عَدِيِّ بْنِ
حَاتِمٍ، - رضى الله عنه - قَالَ لَمَّا نَزَلَتْ { حَتَّى يَتَبَيَّنَ
لَكُمُ الْخَيْطُ الأَبْيَضُ مِنَ الْخَيْطِ الأَسْوَدِ مِنَ
الْفَجْرِ} قَالَ لَهُ عَدِيُّ بْنُ حَاتِمٍ يَا رَسُولَ اللَّهِ
إِنِّي أَجْعَلُ تَحْتَ وِسَادَتِي عِقَالَيْنِ عِقَالاً أَبْيَضَ
وَعِقَالاً أَسْوَدَ أَعْرِفُ اللَّيْلَ مِنَ النَّهَارِ فَقَالَ رَسُولُ
اللَّهِ صلى الله عليه وسلم " إِنَّ وِسَادَتَكَ لَعَرِيضٌ إِنَّمَا
هُوَ سَوَادُ اللَّيْلِ وَبَيَاضُ النَّهَارِ " .
