There is no way of doing that in a standard layout (next to the field). You would need either to create your own visualforce page to see the record and its fields, or add a component to the record's standard layout, which would show the field changes (those can be tracked by a trigger, for example).
If you don't need to do this right next to the field, you can turn on "Track Field History". You'll get a related list-like component on the record page that shows the changes made to the fields (the list won't be turned on automatically if your object is already created. It will be available in the layout editor page. Also, the fields won't be tracked automatically either. In the object's page there will be a Set History Tracking
button for you to select which fields are going to be tracked).