The photo caption about "clean souls and clean feet" is simply a saying that reflects the piety of those who make sure to have clean feet or clean shoes, depending on the practice of the parish, when entering the nave of the Church for worship as prostrations are prescribed in the Russian Old Rite even on Sundays and Feastdays. It would be terribly rude to enter the nave with dirty feet or shoes and make the floor of the nave dirty. Old Believers use podruchniki, prayer mats, that are for keeping the Old Rite worshipper's hands and head from actually touching the floor when making a prostration but still reverence and courtesy are called for when coming to worship and that includes having clean feet.
To answer the question about whether ablution immediately preceding prayer/worship is a rule or religious law as it is amongst Muslims? No, it is not like that at all.