[...]

When a payment is routed through a payment channel, it moves the liquidity from the sender’s side to the receiver’s side. If lots of payments are routed through the same channel in the same direction, outbound liquidity will be depleted, and no further payments can be routed. 

Nodes that have traffic flowing constantly in one direction often opt for a circular rebalancing, which involves making a payment to that channel to push liquidity from one of its channels to another.

While circular rebalancing does solve the problem, it has its drawbacks. 

• Fees can be expensive
• Your other channels are also unbalanced
• Bringing capital on-chain takes time and is expensive

Since liquidity is only flowing one way, the proposal of negative fees would allow fees to flow in the opposite direction. If instead of charging a user to forward a payment, you pay the user and send that additional funds on with the payment. Those fee payments can rebalance your Lightning channel if enough users wish to take advantage of the situation. 

Negative fees are yet to be a feature of the Lighting Network, but one of many proposals to help address channel management, the other being rate cards.

...read more at thebitcoinmanual.com