A buy-back clause lets the selling club re-sign a player at a price agreed at the time of the sale, usually within a set window of years. Big clubs use it when selling a young player they may want back, so a rival cannot price them out later.
It is mostly an academy tool. A club with more talented teenagers than places will sell one to a smaller side for regular football, and write in the right to buy them back if they develop into the player it hoped for. The buying club gets a real signing at a real price and accepts the ceiling that comes with it.
The terms usually restrict when it can be used. A buy-back might be exercisable only in a particular summer, or only once, or only before the player turns a certain age, and the price often rises each year it goes unused.
The clause can also work against the buying club in the market. A player known to carry a buy-back is harder to sell to anyone else, because a third club would be buying an asset another club can reclaim, so some deals attach a higher price to buying the clause out.