Parse literally: 敬 (respect) + 老 (old) + 院 (courtyard/institution) = "respect-the-elderly institution." The name itself reflects the Confucian value of 孝 (filial piety).
culture
Traditionally Chinese elders lived with adult children, but urbanization and the one-child policy have made 敬老院 increasingly common. Also called 养老院 (yǎnglǎoyuàn).