In my eagerness to write about Hamilton, I missed that the core of the question was the word "competitive". As @T.E.D points out, the use of that word is doubtless influenced by meter and rhyme. I would have to check to see if the playwright's use is supported by contemporary language, but the meaning is that those who have capital and want to lend it are looking for the best deal. Primarily, they want to get their capital back, and with the best mix of risk and reward. One can view this as a competition for the capital. The Articles of Confederation were not competitive credit risks - there was strong evidence that they couldn't even make the interest payments. After Hamilton's reforms, there was enough evidence that the USA was a competitive potential creditor.
At the end of the Revolutionary war the United States had a massive war debt (If I recall correctly, it was higher than any time other than the present, or possibly WWII). The new countries primary trade partner was Great Britain, who was reluctant to trade with a country that had a huge war debt to Great Britain and no practical way to pay that debt.
The new country was also a Republic - a type of government that was effectively untested (my Dutch ancestors force me to qualify the statement), and there was widespread doubt that the government could be effective - the rest of the world employed some form of Monarchy to maintain order and property rights, and the notion of giving a bunch of debtors the right to vote themselves money from the public purse was perceived as quite insane.
Aside: How creditworthy is a republic?
technically most European and colonial municipal governments were non independent republics. There were a number of semi independent republics in Europe at the time - Venice, Genoa, Swiss cantons, imperial free cities, etc., so a republic was not exactly a totally untested form of government in Europe. @magolding in a comment
Excellent point - I believe that the tax collection powers of a sovereign republic (vice the subordinate republics you mention) affect the competitiveness of credit. Development of that thesis in the context of your observation would, I think, be interesting, but outside the scope of OP's question.
The new country rapidly demonstrated a thorough and complete inability to collect taxes. Any state could veto any tax collection, or could vote yes and simply fail to provide any money. Functionally the Articles of Confederation was what we would now call a failed state.
Given those things, who in their right mind would lend to the new country?
Hamilton consolidated the debt and organized the revenue service to begin to make payments on the debt. He created a national bank to stabilize the money supply. American credit had been a bad junk bond; after Hamilton, our was competitive with other countries. We were still a huge credit risk, but Hamilton raised our credit rating to the point where we could borrow.
