What do the writers of this petition mean when they claim that the spoils system is “injurious to political parties” and “fruitful of corruption”?

A spoils system rewards those who are aligned with a political party by awarding them some form of public office. This of course is a negative practice to political parties or "injurious" because although someone may be loyal, that does not mean they are competent enough to hold an office, in which an inexperienced person may make poor decisions against the party or poor decisions that are then associated with the party. And also benefits those who may participate in some form of corruption by giving money to get those positions and/or then giving favor once in those positions to the people or party who helped them get those positions, making decisions because they are told to and not because it is in the best interests of those involved.