A signing-on fee is a lump sum paid to the player for agreeing to join a club, separate from the transfer fee paid to their old club and from their wages. It is often split into instalments across the length of the contract rather than handed over on day one.
Splitting it into instalments is deliberate. Paying a fifth of the signing-on fee each year of a five year contract keeps the player financially tied to seeing the deal out, and gives the club something to withhold if the player forces a move.
The fee grows in importance the cheaper the transfer is. On a free transfer there is no fee to the selling club, so the money that would have gone there tends to reappear as a signing-on payment and higher wages, which is why free transfers are rarely as cheap as they look.
It also interacts with loyalty bonuses, which work the other way around: a payment for staying rather than for arriving. A player who hands in a transfer request can forfeit those, which is one of the few real costs attached to asking to leave.