Visitors are normally asked to remove their shoes before entering a Japanese-style home.
Literal
Visitors [topic-は] normally Japanese-style [possessive-の] house [into-に] enter before, shoes [object-を] remove-[indirect-command-ようにと] asked-[polite-passive-られます]
The ~ようにと求める pattern is a reported indirect command — 'ask someone to do X'. Here in the polite passive: 'are asked to'. 日本式 = Japanese-style (as opposed to 洋式 Western-style), a real distinction in Japanese architecture, toilets, rooms, etc. 日本式の家 specifically evokes the removing-shoes-at-the-genkan convention.