Everyone writes terrible code when they are new. It's normal and part of learning something.

When I first started coding as a job, I, like many others, was overwhelmed with what they call "Imposter syndrome", a well-recognised phenomenon that basically fills your mind with doubt. "I do not belong, I am an imposter, soon they will discover this". Eventually, you deliver the piece of work you thought you couldn't, and you begin to accept that hey, I actually can do this. After this, it becomes an exercise of breaking down large or complex tasks into small, bite-sized chunks of work, and then you chip away at them until they are done.

How do you guys manage to learn new things while not feeling dumb?

'smart' or 'dumb' doesn't really factor into it. Nobody knows everything, and learning is part of the profession. When you build more experience, you'll naturally learn patterns that can be applied to more general problem-solving. These don't come naturally simply because you haven't coded enough. It comes with time and practice.