American rebels still had a degree of respect for Britain. Many would still consider themselves British during the Revolution and even after. The war was fought primarily for economic liberty due to aggressive British taxation without any representation in parliament.

After the colonies successfully seceded, the plan was to reestablish economic ties with Britain and possibly become an ally. Washington would even vow that America would never attack the British Isles.

Symbolically speaking, the Red represents bravery and sacrifice, the Blue represents justice and vigilance, and the white represents peace and pure intentions.

A combination of Red, White, and Blue is also simply the most common arrangement of colors in flags and heraldry

When the war began, colonists were fighting to protect their constitutional rights as they understood them. Various scholars, including Brendan McConville and Jack Greene, argue that American colonists held onto an older constitutional understanding in which sovereignty rested with the Crown, while in Britain, political thought following the Glorious Revolution evolved separately toward greater emphasis on the Crown-in-Parliament. An imperial crisis emerged in the 1760s when British administrations attempted to standardize and centralize colonial government through parliamentary means (in America as well as in other parts of the empire such as Ireland).

Historians disagree on the exact moment when colonists shifted their goal to independence (Thomas Slaughter suggests it was when British troops took up arms against colonials in the Battle of Breed's Hill/Bunker Hill in June 1775; Pauline Maier argues it was the publication of Common Sense in January 1776), but it is clear that the general shift in public opinion took time to occur. Here is an example of a paper bill of credit from 1776 where you can see that process playing out. The flag pictured has a Union Flag in the corner and the 13 stripes that would become part of the American flag.

Keep in mind that the revolution was originally British colonists fighting for what they believed were their rights as British citizens.

This is because the colors themselves have meanings different to different nations.

The Union Jack is a combination of several flags, thus fully representing those in Great Britain (apart from Wales, oddly enough). It is the St. George's Cross (ENG), St. Andrew's Cross (SCT), and St. Patrick's Cross (IRE). Combining these 3 creates your hodge-podgy British flag.

The colonists, however, designed their flag differently. The chosen colors have poetic meaning, whether innate or not. White signifies purity, red signifies valor, and blue signifies justice and/or perseverance.

When these colors were being used in pre-enlightenment European flags, little meaning was attributed to color. Interestingly, though, the unified French flag is also red, white and blue. In their unique case, the colors represent either the Virgin Mary or the class makeup of nation they destroyed (White: clergy. Red: Nobility. Blue: bourgeoisie).

So, while the colors are identical to arbitrarily decided European flags, the colonists gave their own meaning to Old Glory. Betsy Ross worked for an upholsterer that did make British flags prior to the revolution. So, perhaps the fabric was easily at hand. Also, the points of the stars were originally suggested to have six points until Betsy convinced them that five points were much easier to sew. Virtually all of the Betsy Ross story is speculative though, as the story was secondarily told by a grandson almost 100 years later.

At this point, any combination of this American mythology and actual evidence can be combined to answer your question.