It is time for opposition to the Annexation of Texas to cease.... Texas is now ours.... Let their reception into the "family" be frank, kindly, and cheerful (O)ther nations have undertaken ... hostile interference against us, ... h pering our power, limiting our greatness and checking the fulfillment of our manifest destiny to overspread the continent allotted by Providence (God) for the free development of our yearly multiplying millions.

California will, probably, next fall away from (Mexico).... Imbecile and distracted, Mexico never can exert any real governmental authority over such a country... The Anglo-Saxon foot is already on (California's) borders.... ?
armed with the plough and the rifle, and marking its trail with schools and colleges. courts and representative halls, mills and meeting-houses.... All this (will happen) in the natural flow of events...

In your own words, what does O'Sullivan mean when he says it is "our manifest destiny to overspread the continent allotted by Providence"?

O'Sullivan is suggesting that it is the destiny of the United States to expand its borders and influence across the continent, as it has been ordained by God. He believes that the US should take advantage of the opportunities presented by the annexation of Texas and the potential fall away of California from Mexico, in order to fulfill this destiny. He sees the US as a nation of settlers, armed with the tools of progress, who will bring education, justice, industry, and religion to the land they inhabit.