For finding answer, other Qur’anic verses along with hadiths (narrations) of Prophet help us.
As for what the straight path and those who follow it are there are two relevant Qur'anic verses:
وَأَنْ اعْبُدُونِي هَذَا صِرَاطٌ مُّسْتَقِيمٌ
and that you worship me? Surely, that is the straight path. Ya-seen
This identifies the straight path with worship of Allah. But as for the "gifted ones" who exemplify this criterion, the following verse (4:69)
وَمَن يُطِعِ اللّهَ وَالرَّسُولَ فَأُوْلَئِكَ مَعَ الَّذِينَ أَنْعَمَ
اللّهُ عَلَيْهِم مِّنَ النَّبِيِّينَ وَالصِّدِّيقِينَ وَالشُّهَدَاء
وَالصَّالِحِينَ وَحَسُنَ أُولَئِكَ رَفِيقًا
whosoever obeys Allah, and the Messenger, they are with those whom
Allah has gifted, the prophets, the sincere, the martyrs and the
righteous, and these are the best company.
Specifies the prophets, the sincere, the martyrs and the righteous whom obey the Allah and his prophets orders, as those gifted by Allah. So these two verses help us, first, define the straight path and, second, identify various groups who qualify as being on the straight path.
Moreover, there are traditions form the Holy Prophet where he clearly name himself and Ali ibn Abi Talib as incarnating the straight path. These traditions imply that Prophet and his most beloved disciple Ali Ibn Abi Talib are the concrete criterion for determining who is truly on the straight path or not. That is, it would be alliance or antagonism with these two figures that should be taken as a reference point for telling who is and who is not on the straight path.
The most notable example of such traditions are found in hadiths that narrate the event of Ghadir Khom, where Prophet declared Ali as his rightful successor, for example:
مَعاشِرَالنّاسِ، أَنَا صِراطُ الله الْمُسْتَقیمُ الَّذی أَمَرَکُمْ
بِاتِّباعِهِ، ثُمَّ عَلِی مِنْ بَعْدی. ثُمَّ وُلْدی مِنْ صُلْبِهِ
أَئِمَّةُ (الْهُدی)، یَهْدونَ إِلَی الْحَقِّ وَ بِهِ یَعْدِلونَ.
ثُمَّ قَرَأَ: «بِسْمِ الله الرَّحْمانِ الرَّحیمِ الْحَمْدُلِلَّهِ رَبِ
الْعالَمینَ...» إِلی آخِرِها
O people! I am the Straight Path of God, whom He has ordained you to follow; and after me, it is Ali, and then my discordance, who shall be
the Leaders of the Guidance and will guide you to the righteousness
and beauty.” Then the Messenger of God announced: ‘In the name of
Allah, (Who's) the most Merciful Bountiful Praise be to God only Lord
of the worlds, The Source of Grace, Mercy, Fair, and the Force, Ruler
of the Day when Repayments begin, Owner of the Day when the Decree’s
given! Alone You we worship, You only the Wise, From You the Sublime
we ask for the rise, Lead us into Direct Path of all the roads, The
Path of the Right, for repayment, You chose, Not of those who were
dishonored, neither those Disowned by Your anger; thrown off the
course!’ (The Opening:1-7)
Additional traditions Tell us that the Holy Prophet, Ali ibn Abi Talib and the infallible Imams from his offspring are the straight path.