I don’t know the specifics of this Buddhist teaching, but there is a lot of scriptural support for fear being closely connected to the practice of killing animals to eat their flesh. Habakkuk 2:17 and Genesis 9:2 link animal fear to the violence of us killing them. Also fear is of the fall. The first thing Adam and Eve did after sinning was hide themselves and their bodies out of fear (Genesis 3), which is the first time fear was introduced into the world.
God’s promise is that animals will one day lie down safely (Isaiah 65:25, Hosea 2:18) without fear. These prophecies are linked to us putting down our weapons of violence and instead leading animals the way we have been called to since the beginning – in God’s image of love (Genesis 1:27-31, 1 John 4:16) by childlike faith (Matthew 18:3, Isaiah 11:6). There is no fear in love because perfect love casts out fear (1 John 4:18). Until we return to God’s original purpose of faithful and wise stewardship (Luke 12:42), our violent ways will continue to fill the earth with fear.