Judicial Independence USA
The principle of judicial independence in the USA has its origins in the UK and for the same reasons. Like the UK the US has a separate judiciary which is is independent from other political institutions. In the US this is even more important since it has the role of determining the constitutional acceptability of the laws and actions of president and Congress.
The separation of personnel means that no one in the executive or legislature—works closely with judges, so there is little chance of close connections or pressure. (By contrast, in the UK, the highest court, the Law Lords was until recently placed inside a legislative body, the House of Lords.)
The president cannot determine the appointment of justices alone, but instead nominates, then the Senate accepts or rejects, having the power to ratify. This could prevent the president appointing someone who will not act independently, because they have close connections to the president.
Justices are appointed for life, preventing a threat of removal. President or Congress cannot remove a justice (though if a justice has acted illegally, Congress can remove them through a supermajority). This gives justices the freedom to act regardless of the wishes of the president of the day.
The judicial compensation clause of Article III protects the pay of judges, Salary stating that their pay 'shall not be diminished during their continuance in office'.