I do not see any transaction in the mempool having differing values for the
modified fee fields. Does nobody modify or accelerate their transactions?
How do transactions with modified fees show up in the mempool?
As Pieter wrote in his answer,
modified fee indicates that the node operator instructed his node to treat a transaction as if it had a higher fee. However, I think that you might be asking due to a misconception of how RBF affects transactions.
Most transactions are immutable: once they have been defined, the signatures on the inputs commit to the transaction's exact content and composition. Updating the fee of a transaction would require at least changing some output amounts. However, changing even one byte of the transaction would invalidate the signatures. (The exception is transaction malleability, which seems irrelevant in the context, so I'm going to ignore that here.) Therefore, transactions cannot be updated.
However, BIP-125 Opt-in Full Replace-by-Fee Signaling specifies how to mark transactions as replaceable which causes nodes to allow conflicting transactions with higher fees to supersede them. While the old transaction remains unchanged and valid in that case, the newer transaction is more lucrative to miners and therefore expected to be prioritized.
I can not see any transaction resting in Mempool having different values for Fee and Modified Fee fields?
That's probably because you haven't modified any.
prioritisetransaction RPC exists for miners to signal that particular transactions are paid for out of band, or are the miner's own transactions that should prioritized for block construction over what is implied by their feerate.
If you're not a miner, there is no need to ever use this RPC, and the fee and modified fee will be identical for all transactions.