Keep at bay means to hold someone or something back, keep it at a distance, or stop it from causing harm for a time.
The phrase comes from old hunting language. Bay referred to the deep barking of hounds, and also to the moment when an animal being hunted was forced to turn and face the dogs. In that situation, the animal was not free, but it was still resisting and keeping the threat back. That image helped create the modern meaning of holding danger off.
Today, people use the phrase for many kinds of pressure or trouble, not only physical danger. It often suggests control rather than complete victory.
- Warm clothing can keep the cold at bay.
- He drank coffee to keep sleep at bay.
- The new policy is meant to keep costs at bay.
A useful contrast is this: if you avoid something, you stay away from it. If you keep it at bay, it is still near, but you stop it from taking over.