This is the 1248th most frequent Russian word.
останется
"Will remain" or "will stay."
The word 'останется' indicates that he will remain or stay in a specific place (home) for a purpose.
Он останется дома, чтобы работать над проектом.
He will stay at home to work on the project.
Here, 'останется' suggests that the rain will persist or continue for the given duration (all day).
Дождь, кажется, останется на весь день.
The rain, it seems, will last all day.
In this example, 'останется' means that something (the pie) will be left or remaining after an action (dinner).