Clear thinking is when you keep any bias, especially emotional ones, out of your decision making process. It is the opposite of cloudy thinking which is not thinking things through, usually because you're being too emotional.
Clear thinking is also more like a habit that one has to acquire. It's not instinctual for us to keep strong emotions out of our decision making process. It requires deliberate effort and practice to become a clear thinker. But the benefits are immense.
Clear thinking is when you don't jump to conclusions rampantly. It is when you put yourself at the different sides of your decision-effect sphere -- in the shoes of everyone who would be affected by your decision. It is when you have a sensible and clearly outlined reason for your actions. It is when you get all the information you need and responsibly use them before making a decision.
In one sentence, clear thinking is sensibly considering all that matters before making a decision.
It saves one not just from lots of embarrassing moments and terrible decisions, but also from a mediocre lifestyle. A high quality life depends on high quality decisions -- clear thinking. If you always take the pain to think through your decisions before making them, for some people you might be seen as annoyingly slow at deciding on what to do but in the end those same people will recognize the quality of the decisions you make and respect you for that. They will be more at ease having you make important decisions than having someone else who is superfast at coming to conclusions/decisions.
Clear thinking also enables you to grow intellectually and be better skilled in handling the affairs of life. Since you've formed a habit of thinking things through and learning from everything that happens around you, you will inevitably grow to become a better and wiser person. You will be able to recognize patterns and a lot more cause-and-effect scenarios which will even speed up your decision process and make you very certain of what you are doing.
There is a lot to gain from forcing oneself to be relaxed, unemotional and thorough in thinking than to be fast and error-prone. In the end, one leads to a better life while the other leads to an error-full life.