I very well understand the story of Bali and Vamana but as a common man I would like to understand what exactly does three steps of Vamana (Vishnu avataar) symbolize?
Matsya Purana- Chapter 260 - Shloka 36 gives us the basic meaning of this three steps of Vamana as "Pervading all the universe" .
तथा त्रिविक्रमं वक्ष्ये ब्रह्माण्डक्रमणोल्बणम्।
पादपार्श्वे तथा बाहुमुपरिष्टात् प्रकल्पयेत् ।। 260.36||
Now about the Vamana form of lord striding the three worlds , as if pervading all the universe .
Shri Adi Shankaracharya in his Vishnu Sahasranama Bhasya name 530 - i.e. Trivikrama is also explaning the above symbolism behind these three steps of dwarf avatara Vamana.
530 - Trivikrama -The three-stepped
Vyakhya - The three steps were in the three worlds. The shruti ( Tai.Br. 2.4.6) says :" He stepped (the three worlds ) by his three steps" ; or has walked over the three worlds. The Harivamsha (279-80) says : The sages have named the three worlds as tri ; and as you walked them all thrice you are named Tri (vi) krama. .
According to Pancharatra tradition of Vaishnavism the three steps of Vamana symbolise that Vamana (Vishnu) had gone beyond the three vedas. The Wisdom Library page in its meaning of Trivikrama is explaning this with the ref. of commentary of Parashara Bhatta as follows.
These Vibhavas (eg., Trivikrama) represent the third of the five-fold manifestation of the Supreme Consciousness the Pāñcarātrins believe in. Note: The name Trivikrama is given to Vāmana who grew up and took three strides (krama). Parāśarabhaṭṭa Viṣṇusahasranāma vyākhyā on Name 533. gives a different interpreation by quoting a passage the source of which is not known. This means that Viṣṇu had passed through all the three Vedas or had gone beyond them. The root kram means to walk over, cross over. The word Vikrama cannot therefore mean studying the three Vedas but has gone beyond them, prominent in them.