'anti-religious' is a modern English compound formed from the prefix 'anti-' (from Greek 'antí-' meaning 'against') combined with 'religious' (from Latin 'religiosus', related to 'religio' meaning 'religion'). 

'religious' derives from Latin 'religiosus' (from 'religio'), passed into Old French and then Middle English as 'religion'/'religious'; the prefix 'anti-' has been used in English since the 17th-18th centuries to form compounds meaning 'against', and the compound 'anti-religious' attested from the 19th century onward as a descriptor for opposition to religion. 

Originally a literal compound meaning 'against religion', the term has retained that basic sense but has also taken on political and ideological connotations (e.g., state-sponsored anti-religious policies, anti-clerical movements).