There were five very good reasons to oppose a Bill of Rights at the time.
The first was that the very concept of a Bill of Rights implied, to many thinkers of the revolutionary era, a monarchy. The British concept of a Bill of Rights originated with the Coronation Charter of King Henry I in AD 1100, followed by the Magna Carta of AD 1215 and the English Bill of Rights of 1689. All three documents were concessions, by kings, to the power of the people's lower-ranked leaders or representatives--a promise by a powerful hereditary monarch that he would not choose to use his power in a certain way.
But in the proposed U.S. system, the people themselves--or at least white male landowners of a certain age--could vote for their own representatives, and hold those representatives accountable on a regular basis. This meant that the people had nothing to fear from an unaccountable monarch; if they didn't like the policies their representatives were implementing, so went the theory, then they could choose new representatives to undo the bad policies and write better policies. Why, one might ask, do the people need to be protected from violating their own rights?
The second reason was that the Bill of Rights was used, by Antifederalists, as a rallying point to argue in favor of the pre-constitutional status quo--a confederation of independent states, operating under the glorified treaty that was the Articles of Confederation. Antifederalists no doubt knew that a debate over the content of a Bill of Rights could delay the adoption of the Constitution indefinitely, so initial advocacy for the Bill of Rights was not necessarily made in good faith.
The third was the idea that the Bill of Rights would imply that the federal government's power is otherwise unlimited. Alexander Hamilton argued this point most forcefully in Federalist Paper #84:
I go further, and affirm that bills of rights, in the sense and to the extent in which they are contended for, are not only unnecessary in the proposed Constitution, but would even be dangerous. They would contain various exceptions to powers not granted; and, on this very account, would afford a colorable pretext to claim more than were granted. For why declare that things shall not be done which there is no power to do? Why, for instance, should it be said that the liberty of the press shall not be restrained, when no power is given by which restrictions may be imposed? I will not contend that such a provision would confer a regulating power; but it is evident that it would furnish, to men disposed to usurp, a plausible pretense for claiming that power. They might urge with a semblance of reason, that the Constitution ought not to be charged with the absurdity of providing against the abuse of an authority which was not given, and that the provision against restraining the liberty of the press afforded a clear implication, that a power to prescribe proper regulations concerning it was intended to be vested in the national government. This may serve as a specimen of the numerous handles which would be given to the doctrine of constructive powers, by the indulgence of an injudicious zeal for bills of rights.The fourth reason was that a Bill of Rights would have no practical power; it would have functioned as a mission statement, and there would have been no means by which the legislature could have been forced to adhere to it. The Supreme Court did not assert the power to strike down unconstitutional legislation until 1803, and even state courts were so reticent to enforce their own bills of rights that they had come to be regarded as excuses for legislators to state their political philosophies. This is why Hamilton dismissed such bills of rights as "volumes of those aphorisms ... which would sound much better in a treatise of ethics than in a constitution of government."
And the fifth reason was that the Constitution itself already included statements in defense of specific rights that might have been impacted by the limited federal jurisdiction of the time. Article I, Section 9 of the Constitution, for instance, arguably is a bill of rights of sorts--defending habeas corpus, and prohibiting any policy that would give law enforcement agencies the power to search without a warrant (powers granted under British law by "Writs of Assistance"). And Article VI protects religious freedom to a degree when it states that "no religious Test shall ever be required as a Qualification to any Office or public Trust under the United States." Many of the early American political figures must have found the idea of a more general bill of rights, restricting policy in areas beyond the logical reach of federal law, ridiculous.
But in 1789, James Madison--the chief architect of the original Constitution, and himself initially an opponent of the Bill of Rights--was persuaded by Thomas Jefferson to draft a slate of amendments that would satisfy critics who felt that the Constitution was incomplete without human rights protections. In 1803, the Supreme Court surprised everyone by asserting the power to hold legislation accountable to the Constitution (including, of course, the Bill of Rights). And in 1925, the Supreme Court asserted that the Bill of Rights (by way of the Fourteenth Amendment) applied to state law, too.
Today, the idea of a United States without a Bill of Rights is horrifying. In 1787, it seemed like a pretty good idea. All of this speaks to the power of words--and constitutes proof that even "volumes of aphorisms" and non-binding mission statements can become powerful, if those in power come to recognize them as such.