EDIT: multiBit no longer has a client fee. I've update this response to remove references to it.
MultiBit HD has two types of fees. It has "bitcoin transaction fees" (also known as "miner's fees"), which are voluntarily paid to provide an incentive for miners to confirm your transaction.
It also has a fixed 1000 satoshi "client fee", which is paid to the developers of MultiBit HD to provide an incentive for those developers to continue to maintain the MultiBit software.See here:
https://multibit.org/help/hd0.1/fees.htmlThe Fees panel provides a slider so that you can adjust your default fee setting to better match your day to day use. The default value provides a good starting position so that it is likely that your transaction will get into the next block, as of Summer 2015
A year ago it might still have been possible to send a transaction with 0 (or very low) bitcoin transaction fees and have it confirmed quickly. However, the transaction volume on the bitcoin network has increased significantly over the past year. Bitcoin mining and pool operation have evolved to become very competitive businesses. In order to remain profitable it is necessary for miners and pool operators to maximize their revenue. Since their main (only?) source of revenue is the transaction fees voluntarily paid on transactions, and since the protocol currently limits them to a maximum of 1 megabyte of transactions in the block they are mining, miners and pools will typically choose to confirm the megabyte worth of transactions that are paying the highest fees per byte (leaving the rest of the cheaper transactions for some other miner to mine in some later block).
The following website monitors the blocks and unconfirmed transactions along with the fees being paid and calculates a reasonable estimate for how much fee a transaction needs to pay per byte for fast confirmation:
https://bitcoinfees.21.co/Looking there right
now, you should consider paying about 200 satoshis per byte if you want your transaction confirmed in the next block or two. This amount changes constantly depending on how many transactions are waiting to be confirmed, and how much those transactions are offering in fees. Think of it a bit like an auction where you are using your fees to bid for space in the next block.
The reason you receive funds quickly is because the people sending you those funds are paying a high enough transaction fee when they send you those funds.