I have heard from a Muslim professor that the Prophet (Sallallahu Alaihi wa Sallam) said that "Those who love passionately but are chaste and died, dies as martyrs" . I would like to know if this Hadith is true? What does it mean exactly?

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

The Hadith you mentioned in your question is fabricated as classified by Imam Ibn al-Qayyim in his book 'al-Manar al-Muneef'. Imam Ibn Hajar (Rahimahullah) in his book 'Talkhees al-Habeer' said: 'the Hadith that reads, the one who loves passionately…, is reported from Suwaid bin Saeed al-Hadathan , from Ali bin Mushir from Abu Yahya al-Qattat from Mujahid from Ibn Abbas that the Prophet (Sallallahu Alaihi wa Sallam) said: "The one who loves passionately and be chaste and then dies is a martyr" . The scholars of Hadith blame Suwaid for this Hadith as Ibn Adi mentioned in his book 'al-Kamil'. Imam al-Baihaqi and Ibn Tahir also criticize him fro this Hadith'.
In brief the Hadith is very weak yet it is a fabricated as confirmed by some scholars.
However, if one inclines to someone unintentionally and is patient and avoids Haram. Allah will give him a great reward.
Allah knows best.