The example at top, "I'm very sorry that it didn't meet your standards," means "I'm very sorry that it wasn't good enough for you."
To meet someone's standards means to be good enough for them. Your "standards" are the basic level of quality that you expect and require. Some people have very high standards, while others have lower standards.
Another example of the word "meet" used this way is to "meet (someone's) expectations". This means to be as good as someone expected:
After waiting for so long to get one, I was disappointed to find that the iPhone didn't really meet my expectations.