Is it true that due to the US government shutdown could lead to have airplanes grounded?
Yes, because the TSA agents employed to help with security are technically working for no pay right now (along with many other government paid agencies such as our military personnel as well as services for disability, food stamps and housing aid) and because of that they are quitting their jobs, partaking in 'sick outs' (calling in sick en masse), and the remaining few that are still willing to work are overworked trying to keep things running as smoothly as possible.
People trying to fly to the recent PAX convention were told to either not go or to arrive very early for their flights. One person I know arrived at the local airport at 1:30 in the morning and made it through security in 15 minutes then simply slept in the terminal near the gate as the flight did not leave until 3pm. But by 10am it was taking people up to 2 hours to make it through security and because they were unprepared many people missed flights entirely due to being held up at security.
Eventually this is going to lead to airports simply shutting down and planes being grounded because there's not enough security staff to handle the absolute volume of people trying to get through. In the US with Valentine's Day coming up and people wanting to fly out to meet loved ones many I know are considering ground forms of transportation or just not going because flying is going to be such a pain.