没关系 Méi guānxì literally means "it is not connected/related to it" or "it has nothing to do with it." It is a very commonly used phrase meaning "don't worry about it," "it doesn't matter," "no worries," that kind of thing.
Another example of 关系 in a sentence is 'X 跟 Y 没(有)关系': "X has nothing to do with Y."
The word 关系 (guanxi) means "relationship", but in the way of "relationship points" you get when you buy someone a gift, cook them dinner, or do someone a favor (the contrary is also true, when you call favors you "use" your 关系). That means once someone does a favor to you it creates an unspoken obligation to repay that favor. Once you and someone else build up 关系 you have a relationship.
Considering that 没 is a word of negation, 没关系 actually means "you don't owe me any favors", so, there is no reason to thank.
别担心 literally means “don’t worry” or “don’t be concerned”. You could be concerned or worried for all sorts of reasons (context).
没关系 (in this sort of context) means that there’s no obligation. And thus can imply not to worry about it. (But in other contexts it could mean that things are not connected, unrelated to each other, without any reference to worry).