In that context, "How I Met Your Mother", Barney literally and specifically means put on a suit, with a jacket and tie.
In other contexts, it means 'prepare for an activity by putting on the appropriate clothes, uniform or equipment.'
Rather than grouping "be out" together, it is probably easier to understand if you read it like "[You will be] [out there] [driving safely again]".
By "out there", he means, "on the road". "Out there" could refer to any location, though. You could say, "Get out there and have an adventure!" if you were telling someone to get out of the house and go do something.
As a side note, whoever wrote this driving guide was speaking very informally. There are a lot of grammar mistakes. It's good to practice reading this kind of writing, but don't worry too much if something doesn't make sense.
As another side note, if you say that a person "is out" about something, it means they had a secret, but now everyone knows about it. Therefore, to "be out" means that people know about your secret. You hear this phrase most often when someone who is a homosexual tells people about it for the first time. When I read the title of your post, I thought it would be that kind of question, haha. :)
Best Answer
It means "I will pay for this", as in "I will treat you to this". Usually said in the context of a restaurant meal, or possibly an activity such as going to a movie or concert.