How does the DynamoDb Point in Time Recovery Works ?

The questions you ask are all leading to implementation detail. PITR is a continuous backup solution allowing you to restore to a new table from any point in time over the previous 35 days.

1. How often do the PITR backup scripts run ?

There are no scripts running, its a continuous backup.

2. Will the backup scripts run for every CRUD operation on the table ?

As above

3. It has been mentioned that we could go back to any point in time in the last 35 days, Incremental backup works in such a way that it takes the difference from the previous backup and current table and takes backup. So Won't we lose the initial data from the backups post 35 days ?

There is no data loss at all.

4. How is the backup size per month being calculated and respective cost being calculated ?

DynamoDB charges for PITR based on the size of each DynamoDB table (table data and local secondary indexes) on which it is enabled. DynamoDB monitors the size of your PITR-enabled tables continuously throughout the month to determine your backup charges and continues to bill you until you disable PITR on each table.

All of the information you should need can be found in this video: https://youtu.be/yvBR71D0nAQ?t=2042