This is quite simple. There are a few things to note when you broadcast a transaction.
Firstly the sender can bump the fee himself with some extra coins from his wallet in doing this it won't affect the initial amount you send since the extra amount spent in bumping was from the remaining balance.
Sender can bump fee with Replace By Fee by two options.
- Preserve payment.
- Deduct payment.
With preserve payment, the initial sending value does not change and sender must have some satoshis to pay the extra bump fee.
With deduct payment, the initial sending value will be deducted by extra bump fee. In this case, the sender does not need to have extra satoshi to bump fee but eventually the sending value of that transaction will be smaller a little bit (by deduction for extra fee).
Also, the receiver can bump the fee probably because they want it received faster(confirmed quickly) so they can speed up the confirmation by creating a child transaction from the initial transaction only with a higher fee.
It's Child Pay for Parents, not Replace by Fee.