It’s hard to say exactly from the picture, which was pretty dark, but it looks like it could be your anti-roll bar (also known as a sway bar or anti-sway bar). The anti-roll bar dampens side to side roll movement and helps keep your car stable in cornering or on uneven roads. It’s safe to drive provided you drive very carefully and avoid and fast cornering or off-roading. If your front bar is broken you could have serious handling problems at highway speeds as well, it depends on the car. Basically as long as you drive like the stereotypical grandma (not mine, she got speeding tickets well into her 70s) you should be okay, but get it fixed!
Like so many other things in mechanics you can replace it at home provided you have the tools and space. You have to be able to get under the car and you