What was the goal behind the 9 state approval of the United States constitution?

The Ratification of the Conventions of nine States, shall be sufficient for the Establishment of this Constitution between the States so ratifying the Same.

Article 7 of the US Constitution

A unanimous vote gives every state effective veto power over the whole enterprise risking deadlock over each state's various pet issues. This gives an individual state disproportionate bargaining power to extract concessions from the rest. Under the Articles of Confederation most decisions required unanimous approval by the states and that wasn't working out so well.

The US Constitutional Convention began as an attempt to fix the problems in the Articles of Confederation. But when Rhode Island refused to send delegates that became impossible, a unanimous vote to amend the Articles was impossible. The prospect of Rhode Island also vetoing the US Constitution was very real. New York was also very tepid to the idea, two of their three delegates left early leaving only Alexander Hamilton.

Vetoing the new Constitution meant everyone goes back to the Articles of Confederation, an outcome some states might be willing to risk, and the union stays together... for now.

By making it a two-thirds vote, no state could hold the rest hostage with their individual demands, a big problem under the Articles. Once ratified, the ratifying states could move ahead under the new Constitution while the hold outs would be left out in the cold. While the necessary two-thirds was achieved on June 21, 1788, the Congress of the Confederation waited until March 4, 1789 to begin proceedings under the new Constitution. This allowed time for the holdouts time to join which Virginia and New York swiftly did, though by narrow votes.

As the holdouts shared no borders, the Articles Of Confederation would be effectively dead. The holdouts would be under increasing pressure to ratify to regain the benefits of union with the other states. The alternative would be to become individual countries. While New York or Virginia might consider that, it would put immense pressure on tiny Rhode Island.

However, this almost backfired. Even ratifying the promised Bill Of Rights didn't bring Rhode Island into the fold. It was the US threatening a trade embargo that finally pushed them over the edge, the prospect of having no land-based trade and hostile neighbors on all sides was too much. They voted for ratification on May 29, 1790 by a narrow 34-to-32 vote.

For more about the rocky early history of Rhode Island and the rest of the US, read “Rogue Island”: The last state to ratify the Constitution by Jessie Kratz, the Historian at the National Archives.

The "nine state" mandate was a two-thirds majority (out of 13 states). That would put the Constitution in force, but only for those nine states.

Rhode Island was the smallest state in the Union in both land area, and population. As such, it was a "take it or leave it" state. It was also resistant to joining.But very few founding fathers wanted to hold back just for the sake of Rhode Island.

The other late joiners were Virginia, New York, and North Carolina. The first two was just slower than others, the last had some doubts about joining. North Carolina was not as critical as the other two, being smaller in the number of potential voters, and less central in geography.

The key states by were Virginia, New York, Massachusetts, and Pennsylvania by potential voters and location. The Constitution was, in fact, ratified without the first two, but its hard to see how the U.S. could have gotten along without them.

Your history book is... not very useful.

Let's correct some language first -

Either this concept is not clear in my head or this is worded bad but wouldn't there be no population sanction if Rhode Island had not approved the constitution in the first place?

First off, I think you're looking for popular sanction, not population sanction.

Second, 12 out of 13 states would definitely indicate popular sanction. (or, more precisely a majority of the voters of 12 out of 13 statues - or, since Pennsylvania cheated, a majority of the voters in 11 out of 13 states, plus enough of Philadelphia. (Pennsylvania never alerted non-English speaking voters of the vote; they pretty much just declared ratification without actually bothering to issue or count ballots).

Remember that in the end, Rhode Island refused to ratify the Constitution until the other states threatened to treat Rhode Island as a foreign power - which would have doomed Rhode Island. Rhode Island was pursuing a populist free money policy that was extreme by any standard.

So Rhode Island was irrelevant - your textbook overestimates the importance of Rhode Island. I don't think anyone's calculations included Rhode Island any more than I consider the vote of my pet turtle.

That said, the Constitution needed popular sanction - everyone needed to believe that the Constitution represented the government the people chose.

One of the things that doomed the Articles of Confederation was the requirement for unanimity; the 13 states couldn't have passed a resolution that there were two digits in 13. By the time of the ratification debate, if New York had advanced the notion that water was wet, the South would have voted that water was in fact both flammable and solid, but never wet. The Articles of Confederation had taught the States that the best strategy was to be the last vote cast and that it was wise to vote against your own interest if by doing so you could extract prices from the other States. I can't call the examples to mind, and I can't find my copy of the book, but Klarman offers some examples of strategic voting that were almost as silly as my example of "water is wet".

So the level of popular sanction was set at a majority of the voters in 2/3 of the states. (the Constitution was not submitted to the states, but to the voters, but that is outside the scope of your question.) A majority of the voters in 2/3 of the states was sufficient to ensure that everyone would accept the legitimacy of the resulting government - that's the way "popular sanction" should be interpreted.

Best references are Maier's Ratification and Klarman's Coup. The origin of the Constitution is complicated and fully of wonderfuly juicy politics, but it is difficult to distill down to fit within the requirements of a High School Textbook (which is usually the first introduction to a complex subject, but must be written to contain nothing complicated, objectionable or confusing. They are written to be fairy tales to pass the scrutiny of the Texas School Board, an institution respected more for it's ideological purity than academic standards).