Cromwell's head on a pole outside Westminster was part of his ritual execution and display as a traitor to the realm. I imagine you are inquiring about why it was put out at all, rather than the length of time it was left there.

So, the first item to consider is that Oliver Cromwell died in 1658 of natural causes, and his head was impaled almost three years later in 1661. In the interim, Oliver's son Richard came to power as the new Protector and failed miserably. Charles II was invited to return to England as its King in 1660, and he would arrive to great celebrations in May of 1660. Charles II's first parliament met from April to December 1660, and one of the lasts acts it passed was an act of attainder against the regicides. Part of that act required the corpses of Oliver Cromwell, Henry Ireton, and John Bradshaw be disinterred, drug to Tyburn (the famous location of most of London's executions) and hung in their caskets. The was done on January 30, 1661, and was done in front of an audience of thousands. Within a few days, Cromwell's body was cut down, his head impaled and brought to Westminster. The exact date isn't known, but we do know that Samuel Pepys went and saw Cromwell's spiked head on February 5, 1661.

All of this was done to great fanfare, the bodies being disinterred, drug through the streets, hung and butchered were all part of the carnival of execution, so common in early modern Europe. Cromwell's head remained on a pike for at least 10 years, as it was not recorded when the head was taken down (or why). It remained on the spike as a reminder of what happens to traitors to the crown, usurpers of power, and was spoken in the language of physical violence - a common tongue spoken by most everyone in early modern England.