Well actually I hear more guys say don't trust anyone than women. But I would say because women are very easily taken advantage of when they are vulnerable and naive so typically at a young age. Generally there is some kind of even in our adolescents that just makes us not trust anyone. Like a boyfriend cheating on us or hurting us, our parents betraying us in some way. Basically any emotional, verbal or physical abuse. Of course men experience this too but I think it happens to them when they are older and know better and are more capable of handling the situation.
Trust is a pretty precious thing. When you trust someone, you're opening yourself up to them and you're giving them a lot of power - power to hurt you, betray you, negatively affect your life, destroy your friendships/relationships.
I think trust is a beautiful thing, but the concept behind what they're saying is that if you never trust anyone, you can't give anyone the opportunity to abuse it and hurt you.