The US Government has $\\\$X$ in spending and raises $\\\$Y$ in revenue.
Usually, $X>Y$. So, the US Government (or more specifically the US Treasury) has to cover this $\\\$(X-Y)$ difference (or funding requirement/shortfall).
It does so by issuing bonds (or notes or bills) worth $\\\$(X-Y)$. Pretty much anyone is free to buy these bonds, including the public and the Fed.
So "writes a check to Congress in exchange for them" probably simply describes the act of the Fed buying some of these bonds.
The buying and selling of US Government bonds by the Fed is called open market operations. See e.g. Investopedia or this Fed webpage for how this works.
- The US Congress doesn't have any direct official/technical/legal role in any of this, except insofar as it determines spending ($X$) and revenue ($Y$). So, it's not really correct to say that the Fed "writes a check" to Congress. We should instead say that Fed "writes a check" to Treasury. (Of course, we could quibble and say that Congress and the Treasury are all part of the US Government, so "writing a check" to the Treasury is "really the same thing" as to Congress.)
- This writer writes, "the Fed takes all the government bonds which the public does not buy." This statement may suggest incorrectly that the government is free to sell as many bonds at whatever interest rate it likes. But this is incorrect. Instead, what happens is that the Fed targets certain interest rates. The government must then adjust its interest rates—so as to meet its $\\\$(X-Y)$ funding requirement and, at the same time, the Fed's targets.
- "Writes a check" is not strictly speaking correct, since nowadays this is all done electronically with keystrokes rather than any physical writing of checks.
By the way, I've never heard of this writer, but Wikipedia calls him a conspiracy theorist and states that The Creature from Jekyll Island (1994) "promotes false theories about the motives behind the creation of the Federal Reserve System." So I wouldn't necessarily believe anything written by this writer.