What is the ruling of taking the fruits from the tree of my neighbour that has fallen into my backyard and eating them?
Basically, taking the rights of another without permission is prohibited. Allah SWT states:
وَلاَ تَأْكُلُواْ أَمْوَالَكُم بَيْنَكُم بِالْبَاطِلِ
“O ye who believe, do not consume your wealth among yourselves unjustly,” 
Likewise, it is also stated in another hadith of the Prophet PBUH:
لاَ يَحِلُّ مَالُ امْرِئٍ مُسْلِمٍ إِلاَّ بِطِيْبِ نَفْسٍ مِنْهُ
“A man’s property may not be taken except with his goodwill.” 
Syafi’iyyah scholars explained as cited by Imam al-Nawawi, the ruling of fruits that have fallen is the same as the ruling of fruits when they are still on the tree. This means that it is prohibited to take and eat them without asking for permission. If the fruits fell outside the gate, it is still prohibited to take and eat them without the permission of the owner except if it is common knowledge that customarily the owner of the tree allow others to take the fruits or there is an assumption that the owner allows it. 
However, we should also celebrate the customs of our people where we respect our neighbour and would usually first inform the owner before taking the fruits. It would calm our hearts and avoid any misunderstandings in the future. Wallahu a’lam.