What problems with the original document, or changes in society, led to later amendments?
I am assuming you are referring to the amendments after the Bill of Rights. There were many problems and societal forces that required changes. I will discuss just two.
One set of circumstances that led to the 14th and 15th Amendments was the aftermath of the Civil War. Bear in mind that the Bill of Rights guarantees that the federal government must ensure the rights contained therein. There is nothing in the Constitution or the Bill of Rights that promises the states will preserve any rights at all. After the Civil War, it became apparent that the Southern states were going to do everything possible to make life difficult for the slaves who were now freed. So the 14th Amendment makes clear in its language that the states must provide all the "privileges or immunities" of citizenship, and the 15th makes clear that race may not affect these rights. Without this language, the states would have been free to do as they pleased.
An interesting circumstance led to the 22nd Amendment, the administration of Franklin Delano Roosevelt. Roosevelt was elected to office four times! There was a feeling that this was too long for anyone to be president, so the 22nd imposed term limits on the presidency. Roosevelt had died before the ratification in 1951, so he had no opportunity to be elected a fifth time, but the principle is a good one. Too many years of one person at the helm is probably not a good idea.
For most of the amendments following the Bill of Rights, there is a story. What you might notice as you examine them is that in many cases they were reactive, rather than proactive, ways of making adjustments to situations the framers had not contemplated or responding to changes in society. Even today, most of the discussion about any new amendments is quite reactive, for example, the discussion about an amendment to require marriage be only between a male and female. This is clearly a reaction to the assertion of equal rights by gay people, creating a backlash among various groups who believe that homosexuality is "wrong."