to take refuge: to flee to someone or something for shelter and protection
This verb pictures a person running to a safe place and putting themselves under its protection. It is the act of someone exposed and in danger who shelters behind a stronger party and counts on being kept safe there.
Far and away its most common use is taking refuge in God. It is a favorite word of the Psalms, where the worshiper flees to the LORD the way a hunted person runs to a fortress or a small bird tucks under a larger bird's wing. The same picture can be turned on its head: fleeing to a physical shelter like the shade of a tree, or worse, fleeing to the wrong protector, like trusting a foreign army or an idol to keep you safe.
What ties all of these together is the picture of seeking cover, not the thing you run to. The Bible keeps the picture and changes the object to make its point: shelter taken in God holds, while shelter taken in anything else fails when the danger comes.
It overlaps with the more common Hebrew word for trust, but it is more concrete. To trust is an inner confidence; to take refuge is to actually run to a hiding place and stay there. That is why God is so often called a rock, a shield, or a fortress in the verses that use this word.