Why did the Articles of Confederation fail, and how was the Constitution supposed to be a better choice?

The Articles of Confederation had helped the United States win its war with the British, but it couldn't help win the peace. The deliberately weak system of government established under the Articles proved woefully inadequate to deal with the challenges of a new nation emerging after years of bitter conflict.
The animating spirit behind the Articles was a radical republicanism which distrusted any kind of strong, centralized government. The American colonists had just fought a war against what they regarded as a tyrannical central government, and the last thing they wanted to see was its reintroduction by the back door. So under the Articles of Confederation, ultimate sovereignty resided with the states.
However, this generated a number of serious practical problems, which the proposed new Constitution set out to address. For one thing, without a centralized government in place, the United States was unable to pay off the enormous debts it had accrued during the Revolutionary War. This damaged the American economy and meant that other countries wouldn't be able to trust the United States as a reliable trading partner. Supporters of the new Constitution believed that only the establishment of a strong Federal government and a Federal banking system could remedy the situation.
The Articles had also proved to be completely incapable of dealing with domestic disturbance. As there was no Federal authority to speak of, outbreaks of disorder could only be dealt with by the individual states, but if the states were unable or unwilling to act, then there was nothing that the weak Congress could do to prevent such disorder from spreading beyond state boundaries to engulf the whole country. Indeed, one such serious outbreak, Shay's Rebellion, was the catalyst behind the convening of the Philadelphia Convention. A strong Federal government was required to maintain peace and stability in the United States, and the proposed Constitution set out to achieve this.