Depends what theatre of the conflict you're referring to, because it was really two separate wars that happened to share a couple of the same belligerents (Britain and France).

The colonial and overseas conflict between Britain and France (usually referred to as the French and Indian War, in North America) began in 1755 following disputes regarding the construction of forts along the Mississippi River. The French claimed ownership of the entire river and both of its banks, but the British disputed this, primarily because it would heavily restrict British traders from operating there and being the main source of goods for Native American tribes. The French wanted to exclude the British from trading in the interior of North America, but since French goods were both more expensive and of a lower quality, they couldn't let the free market do its work; Native Americans simply preferred British goods. By claiming the land as French land, the French would be able to legally expel British traders and essentially monopolise all trade along the river.

A small force of British colonial (or 'provincial') troops under the command of a young George Washington were sent into the disputed territory with orders to request the removal of French troops and the dismantling of their forts. The French refused, and a brief 'battle' occurred (really a small skirmish), in which the British lost. The situation escalated from there, with neither country's home government willing to back down. Several other skirmishes and small battles occurred along the borders of the Thirteen Colonies and New France (i.e. Canada and the Mid-West), although war was not officially declared until 1757, almost a full two years later.

At first both sides merely wanted to defend their territory and enforce the status quo, but as the conflict escalated British ambitions rose to eventually conquering all of Canada. France, with a colonial population of only 60,000 compared to the British's 950,000, could not realistically hope to conquer new territory in offensive actions, and merely planned on defending what they had (which, in terms of territorial extent, was already vastly larger than the Thirteen Colonies, if much more sparsely populated).

An additional factor that came into play on the more local level was that most Native American tribes in North America were allied, or at least on friendly terms, with the French, and the French would occasionally lead bloody and destructive raids on British settlements. For years the settlers along the border regions had begged the British government to put a stop to these raids, and this new war seemed like the best chance they had to do that.

This colonial war between Britain and France extended across the globe, with other battles and conquests occurring in the Caribbean, West Africa and India. Spain entered the war in 1762 on the side of the French, citing their own many grievances against British colonial actions, such as the British illegally building woodcutting settlements in Honduras (part of the Spanish Empire) as well as their continued occupation of the Spanish island of Minorca (ceded to them following the British victory in the War of the Spanish Succession half a century earlier).

The European theatre saw conflict erupt in 1756 when Frederick the Great, the King of Prussia, invaded the Duchy of Saxony, an ally of the Habsburg Monarchy (the formal title for the empire headed by the Habsburgs of Austria). He did this because he knew war coming; 15 years earlier he had conquered Silesia in the First Silesian War (1740), a large and prosperous province of the Habsburg Monarchy. Though he won the war and held onto the province, everybody knew that it wasn't the end, and for years the Habsburgs had been planning on retaking it. Empress Maria Theresa of Austria formed an alliance with Russia in 1755; the Russians had their own territorial ambitions in Eastern Prussia, and the Russian empress, Elizabeth, personally hated Frederick almost as much as Maria Theresa did (Frederick had a habit of insulting female rulers quite harshly). Frederick initially had an alliance with France, but this broke down after Frederick repeatedly broke treaty agreements and dishonoured the alliance. Also, unsurprisingly, the King of France also hated Frederick because the latter had insulted both him and his mistress. King Louis XV was obsessed with personal honour and viewed Frederick as a deeply dishonourable man that couldn't be trusted.

Frederick was completely isolated by 1756, surrounded on all sides by hostile neighbours that wanted to seize territory and humble his ambitions - the Russians to the east, the Austrians to the south, the French to the west, and even the Swedes to the north. A last desperate attempt was made to form an alliance with Britain; initially the British despised and feared Frederick as much as everyone else did, since it was widely believed he had ambitions on conquering Hanover, a possession of King George II of Britain. The British were even about to subsidise a Russian army poised to invade Prussia. However, when war broke out with France in North America, the British now feared a French invasion of Hanover more than a Prussian one. Frederick saw his chance, and promised the British he would not just leave Hanover alone, but would even assist it in defending against the French, if the British formed an alliance with him and gave him huge cash subsidies. Under pressure from George II to defend his Hanoverian homeland, the British government reluctantly agreed to Frederick's proposal, and so the set of alliances was formed.

Frederick knew war was coming, and he was determined to strike first. He realised that he couldn't take on the full might of all of his enemies at once, so he hoped that by quickly taking out Austria first he could fracture the coalition before it could act in unison. He started by invading Saxony, an Austrian ally, and in an early preview of the Blitzkrieg he conquered the entire duchy in a week. He proceeded to ruthlessly exploit the duchy for all its wealth and resources to fuel the Prussian war effort. This nakedly aggressive act shocked the French, who decided to increase their contribution to the war by sending a huge army of over 100,000 to invade Germany. However, their primary goal was the conquest of Hanover, to use as a bargaining chip in the inevitable peace talks with Britain. The French realised that they weren't going to be able to protect their overseas colonies, and so they needed something of Britain's to balance things out. The British sent thousands of troops to fight the French in north-western Germany, in conjunction with Hanover's army, which was put under the command of Frederick the Great's brother-in-law, Duke Ferdinand of Brunswick, who was answerable to the British government and not Frederick. This is how the two separate conflicts overlapped, despite the massive unpopularity of the war in Germany with the British public and most MPs - the Prime Minister, William Pitt, preferred to focus entirely on the colonial war, but pressure from the king forced his government to become involved on the Continent as well.

Sources:

The Seven Years War in Europe by Franz A. J. Szabo

The Global Seven Years War by Daniel Baugh