I have added a lightning component to custom object page, which takes the data from DB and displays data in a table. When I edit a Custom Object record and click on save I expected my component also refreshes with a page but it doesn't and shows old data. any inputs?
It will not happen until and unless you have some kind of functionality written in your component which will check for change in the data (which is edit here). I will suggest you to go through this documentation to understand what you can do when there is a record change.
Basically you need to fire some event and then in your component, catch the event and perform your business logic. This is known as pub-sub pattern.