You'd have to read yourself. You're not a 5th grader. Or are you?
In addition to the plain language, it was created by Congress. Its governors are appointed by POTUS and confirmed by the Senate. Its independence doesn't preclude it from being a government agency, no more than the Department of Justice.
Colloquially, people often use "government" to specifically refer to the executive branch. When we discuss government as opposed to the free market, we use "government" in a more general sense, which includes all 3 branches of government.
As an aside, in a landmark case Shelley vs Kraemer, the Supreme Court ruled that enforcing discriminatory private covenants constitutes state action under the Fourteen Amendment. In other words, our judiciary is part of government. Shelley vs Kraemer, though not well known, is the seminal case of desegregation cases which culminated in Brown vs. Bd of Education. Robert Bork's opposition to Shelley vs Kraemer turned a lot of conservatives against him. That IMHO ultimately sank his nomination.