You have to fill a physical form that's available on their website and submit it.
You'll receive the following sms on your registered number after successful activation.
"Your India Post Internet/Mobile Banking UserID is ready for activation. Please activate using 'New User Activation' link in https://ebanking.indiapost.gov.in".
It's not quite so vital these days. They have the necessary information on their systems.
I have not needed my passbook for any specific transaction.
That being said, it's never a good idea to lose it. Post Office instructions and processes widely differ in practice, depending on where the PO is located. Perhaps in a rural area, or even the suburbs of a city or township, they may absolutely need the passbook.
For transactions, you can use the debit card, or MB/NB, or a chequebook. If you wish to withdraw cash at the Post Office without any of these instruments, or even with a self cheque, they may ask for a passbook, as they would in several others. Similarly for FD, RD, insurance etc.