This practice is no doubt hygienic especially so if you have a baby crawling on the floor and carpets but is there anything in Qur'an or Sunnah? We take our shoes off for the mosque so is this practice carried across?
In the time of the Prophet, shoes would be worn while praying, but remember the Prophet's mosque was made of earth... The Prophet did take off his shoes to pray as well, once during the prayer when Jibreel told him there was some dirt on it. We should however revive the Sunnah of praying in shoes, not inside the mosque or the home as you mentioned correctly that it is more hygienic to take off the shoes especially in the polluted environments we find ourselves today. But we should revive praying in shoes where possible.
Regarding generally removing the shoes in homes, it is an issue of cleanliness which is part of faith.
A beautiful hadith which I quote in full even though only the first few words relate to this issue:
عَنْ أَبِي مَالِكٍ الأَشْعَرِيِّ، قَالَ قَالَ رَسُولُ اللَّهِ صلى الله عليه وسلم " الطُّهُورُ شَطْرُ الإِيمَانِ وَالْحَمْدُ لِلَّهِ تَمْلأُ الْمِيزَانَ . وَسُبْحَانَ اللَّهِ وَالْحَمْدُ لِلَّهِ تَمْلآنِ - أَوْ تَمْلأُ - مَا بَيْنَ السَّمَوَاتِ وَالأَرْضِ وَالصَّلاَةُ نُورٌ وَالصَّدَقَةُ بُرْهَانٌ وَالصَّبْرُ ضِيَاءٌ وَالْقُرْآنُ حُجَّةٌ لَكَ أَوْ عَلَيْكَ كُلُّ النَّاسِ يَغْدُو فَبَائِعٌ نَفْسَهُ فَمُعْتِقُهَا أَوْ مُوبِقُهَا " Abu Malik at-Ash'ari reported: The Messenger of Allah (ﷺ) said: "Cleanliness is half of faith and al-Hamdu Lillah (Praise be to Allah) fills the scale, and Subhan Allah (Glory be to Allah) and al-Hamdu Lillah (Praise be to Allah) fill up what is between the heavens and the earth, and prayer is a light, and charity is proof (of one's faith) and endurance is a brightness and the Qur'an is a proof on your behalf or against you. All men go out early in the morning and sell themselves, thereby setting themselves free or destroying themselves." (Sahih Muslim).