I've read that he did so because of British pressures, but I'm curious to know what exactly happened, and how it developed to that point. Was the UK threatening war? Was Louis-Phiippe booed at home for having his son Louis of Orleans, Duke of Nemours, relinquishing the crown that was offered to him?
The situation in Europe in the fall of 1830 was extremely tense. France had overthrown its king, Charles X, in the July Revolution, and replaced him with Charles' more liberal cousin Louis-Philippe. This was a problem because after the downfall of Napoleon, the other great powers of Europe had formed a Quadruple Alliance for the express purpose of preventing another destabilizing revolution in Europe, and especially in France. Now, just 15 years after Waterloo, another Revolution in France had toppled the dynasty that had been placed on the throne by these same European powers. Some of the same individuals responsible for enthroning the Bourbons, most notably the Austrian statesman Metternich, were the ones in charge of the response to the French revolution of 1830.
Louis-Philippe's first priority as king was to prevent a very real possibility: an armed invasion of France by a European coalition aimed at crushing the July Revolution and putting Charles (or his son or grandson) back on the throne. To head this off, Louis-Philippe sent ambassadors to the great powers of Europe with a simple message: though he had been elevated by a revolution, Louis-Philippe had only taken the throne to head off a more radical republic, that he had a handle on the situation, and that his new regime — unlike the First Republic of living memory — posed no danger to its neighbors.
This was a policy with enemies in France itself. Many Frenchmen still chafed under the humiliating conditions imposed by the Congress of Vienna in 1815, which included territorial concessions. French nationalists believed the country had a right to its "natural frontiers," which chiefly meant a northeastern border along the Rhine River. If you look at a map, you can see that Belgium lies on the French side of the Rhine, and for most of the Revolutionary and Imperial era, it had been incorporated into France. The Congress of Vienna had stripped Belgium from France and incorporated it into the Kingdom of the Netherlands, an attempt to create a strong buffer state on France's northern border.
Fast forward a few months after the July Revolution and the Belgians, partially inspired by the French example (as well as by domestic sentiments, and a poorly chosen opera), launched their own revolution against their Dutch king. The European powers that had reluctantly — sometimes very reluctantly — accepted the July Revolution as the least-bad option now saw what they had explicitly warned about happening: the contagion of revolution was spreading from France to other European countries. Meanwhile French nationalists saw exactly the same thing, except they saw it was a good thing: they would welcome their Belgian brothers into a new, glorious France.
All this posed a serious danger of European war. At a minimum this would involve foreign troops crushing the Belgian revolt and forcibly containing France's northern border; in the worst case scenario an intervention might try to take on the July Monarchy in France as well.
So when Belgian revolutionaries offered the throne of their new country to Louis-Philippe's son, it was a a gift of a poisoned chalice. Accepting would give France de facto control of a strategically vital neighbor. Even without concerns of revolution this might have prompted revolution — a century earlier, the continent-wide War of Spanish Succession had broken out when other European powers refused to accept a French prince inheriting the throne of another neighbor, Spain. Belgium was smaller than Spain, but for both ideological and geopolitical reasons the great powers of Europe weren't willing to see it fall under French dominion.
Louis-Philippe's biographer Munro Price sums up the situation thusly:
Since it was clearly unthinkable that the new state should become a republic, a monarch would have to be chosen. At this point the Belgians, hoping to extort more concessions from the Dutch by committing France irrevocably to their own side, offered the crown to Louis-Philippe's second son, the duc de Nemours. As intended, this initiative placed Louis-Philippe in an extremely difficult position. Refusing the Belgian crown for Nemours would expose him to furious attack from radical and nationalist opinion within France, perhaps even to another insurrection. Accepting it, however, would lead to war with the rest of Europe. With considerable courage, Louis-Philippe turned down the Belgian crown on his son's behalf, and stood firm despite the abuse heapted on him by his domestic enemies.
As part of a compromise, the German prince Leopold of Saxe-Coburg-Gotha was offered the throne. He had served in the Russian army against the France, had previously married a British princess (and was nominally a British subject), but agreed — since his first wife had since died — to marry one of Louis-Philippe's daughters to secure French support for his bid.
The British were the friendliest of the European powers to Louis-Philippe's regime — he had lived in exile in Britain, had many friends there, and besides, the British could hardly frown on a country ejecting their king for a more liberal dynastic alternative when their own Glorious Revolution was such an important part of the national history. But Britain's geopolitical interests were also firmly against a great power like France dominating the ports of the Low Countries; it had spent a generation at war (despite its own immediate security being rarely directly threatened) to prevent French domination of Europe. In fact the Duke of Wellington, who 15 years earlier had led an army into Belgium to thwart the French, was prime minister at the start of the Belgian crisis (though he would fall for largely domestic political reasons before the crisis of the duc de Nemours reached its peak (in February 1831). Historians debate just how certain war would have been if Louis-Philippe had accepted the Belgian throne, but British participation in such a war would have been a definite possibility.
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