Do you think the government's primary function is to operate a laser that keeps the sun from crashing into the country? You see, a community of people come together and say they could do things more efficiently and have a better life if they all cooperate, so they form a "government" and that government does things for the benefit of those people. In a shutdown, some of those things are deemed essentially and keep happening, like law enforcement and coast guard and border patrol and food inspection and air traffic control. The people doing those jobs just don't get paid! Hooray!
Other things stop working, like FCC complaints, or social security checks, or the IRS help line, or parks, and certain people that help with criminal investigations, or the building of semiconductor plants intended to help the US compete with China, or visa applications, or school lunches for students, or the collection of statistical reports on the US economy that busiensses rely on, and so on. When these things don't happen, life for Americans gets worse. You are quite correct that the country will not immediately burst into flames. All that will happen is that millions of people will suffer! What do we even pay the government for, amiright guys?!