Open minded isn't weak minded. Weak minded means one whose opinion changes every single time someone gives advice or criticizes. Open minded means to be open to receive any advice or criticism, evaluate it, weigh it, consider it, and then either dismiss it or adapt one's own mind to include it.
Why are people closed minded? Many reasons, but often because an idea is foreign to them or seems to be the opposite of the way they grew up or what they've always believed. Change is a scary thing to many people because it is uncertain. So to change one's mind or even entertain the idea might very well be terrifying.
But developing certainty not in circumstances but in your own wits and wisdom will mean that you can be open but not thrown off end by new information or opinions. The word, "offended" means to hurt or go against, but how many times are people deliberately offended versus offended just because an opinion is different than theirs? An open mind allows not only tolerance, but acceptance that others can be totally valid in feeling or thinking differently about certain things without being "wrong."
An insecure person with low self confidence is more likely to be closed minded because they don't feel they can handle anything other than what they think they know. They cease to learn. A secure, confident person can be open minded and continue to learn all throughout life.
Develop confidence and keep an open mind. Life will continue to expand and change will be met with calm instead of panic or anxiety.