My interpretation of this legendary quotation from Franklin D. Roosevelt is that it is not the actual thing that you fear which is the problem, but, the emotion and thought fear surrounding which can have a paralyzing effect.
For instance, say that you are afraid of public speaking, which is the most common fear amongst people. The actual act of public speaking when you are engaged in it during the moment is not bad. It's not that scary, you are in the moment and just doing it. It is the anticipation, the thoughts of messing up, the uneasy butterflies in your stomach of thinking something terrible can happen and you will be mocked which causes a sense of panic and anxiety. When you're actually in the moment facing these so-called fearful events, you can do it. You can handle anything in the now. But it's the thought of future and past that is often the destroyer of happiness. And the thought that you won't be able to handle something in the future is what causes this anxiety in the pit of your gut.
Our overactive fear is probably just a survival mechanism which was rampant in our ancient brains and we were constantly in need to protect ourselves from predators and whatnot. It's better to be afraid and imagine the worst just in case, because if we do not then we might be eaten alive. But nowadays as humanity has taken dominion of the planet, there isn't such a strong need for this predatorial fear, but it is still strongly lodged inside of our brain. So paranoia which was a defense mechanism for survival thousands upon thousands of moons ago has now been transferred into modern day society issues like worry about what people will think about us.
At the end of the day, if we are able to accept our fear, and not be afraid of it, it will make our lives better. The way we react to fear is key. More accurately, the we do not react to it. If there is fear, let it be. It comes, it stays, it goes. If there is nervousness, let it come and go. If there is anger, let there be anger, let it come, let it go. With constant practice of this these emotions which totally overtake us will be allowed to just flow through our consciousness like traffic passing the streets. You have no attachment to the cars flowing down the road, and one day you will have no attachment to the fear which once consumed you.