Learn English – Meaning of “course of the year”

Henry's answer is exactly right. Here is confirmation:

A clearer, more complete use of "cutting" comes later in the same paragraph.

He wasn't willing to cut a subsidy for heating bills.

This illustrates that "cutting," as used in your example, had an object (parts of the proposed budget), and was not just used as an expression meaning something else.

The sentence in your question might be paraphrased in this way, for the sake of this explanation: "As the government programs fell and the tens of billions in savings piled up during the budget debate in the House last week, freshman Republican Frank Guinta joined fellow congressmen in reducing spending and removing parts of the proposed budget.

I think it's just a corruption of "I don't care about getting dirty", similar to how "I could care less" is often said when the user obviously means "I couldn't care less."