I'll give the short not so short in hindsight, but pretty bare bones, answer, and an expert can come in and give you the details. I apologize in advance for all the names but if I try and leave things out it might get even more confusing:

Charles I was the king of England till he was beheaded 1649, after which England spent a now somewhat forgotten 11 years as a Noble Republic (ruled by Oliver Cromwell as a kind of aristocratic military dictator).

After those eleven years, the monarchy was restored under Charles I's son, Charles II. However, during their exile the royal family had largely returned to Catholicism, which became a source of strife with the largely protestant English aristocracy. Charles II died without legitimate heirs and was succeeded by his brother, James II (So another son of Charles I).

This is where the trouble starts. At the time of his ascension to the throne in 1685, James II's heir was his daughter, Mary II. And unlike her father, Mary II was a protestant. This of course greatly pleased the protestants in the English aristocracy who wanted a protestant monarch.

Then in 1688, James II's wife gives birth to a son. Since males went first, this baby boy was now first in line of the throne ahead of his already 26-year-old sister Mary II. Since this son would obviously be raised Catholic, this dashed the hopes of the protestant aristocracy who had been expecting a protestant to succeed James II. So they began to scheme a plot to usurp James II and replace him with Mary II.

So, now where does this William III come in?

William III was Mary II's husband. William III was the Prince of Orange and Stadtholder of the Dutch Republic. The internal politics of the Dutch Republic was very complicated, but the quickest way to explain it is that William III was essentially President and Commander-in-Chief for life.

William III had his own reasons for wanting James II off the throne. The Dutch Republic was severely, existentially, threatened by the kingdom of France. In 1672 an alliance of England and France attacked the Dutch Republic with the goal of wiping it off the map, and came within a hair's breadth of succeeding. It was this war which had allowed William III to ascend to his position as de facto ruler of the Republic, but he knew it had been a close call and a continued Anglo-French alliance would possibly be the end of his nation.

The two sides, English protestant aristocrats and William III, found each other and in 1688 William III sailed the Dutch fleet across the channel and landed in England with the Dutch army. With the support of much of the English aristocracy, William III faced little resistance and James II fled the country. William III took control of London and began a series of negotiations with parliament as to what their next step would be.

Now an expert might correct me on the inner details of this negotiation [edit: u/tim_mcdaniel outlined it in more detail in this post], but as far as I know this is the basics: the majority of the English aristocracy wanted Mary II to be the sole queen and ruler of England. But William III opposed this. For one, he also had a dynastic claim to the throne because he was a grandson of Charles I through his mother. For two, it was his Dutch army in London that had ousted the previous king James II. Now in all this, Mary II fiercely supported her husband. All accounts I've read suggested that William and Mary were an unusually loving and supportive couple (unusual considering theirs had been an arranged marriage as all were in that time).

The compromise reached was that both Mary and William were made full and equal monarchs of England. Which afaik in practice meant William III ruled. Note that Mary died eight years before her husband, after which he continued to rule as sole monarch.

During this time Britain and the Netherlands were two nations ruled by the same person. But as William and Mary had no children, after William's death the countries split again. After a brief reign by Mary's sister the English crown fell to the house of Hannover. The Dutch situation was, as always, complicated, but the successors of William's line (and ancestors of the current King of the Netherlands) was a distant branch of the House of Orange which had split at the time of William III's great-grandfather.

Sources: My own main source on William III is Rampjaar 1672 by Luc Panhuysen (in Dutch). But it deals primarily with the war in 1672 and not the Glorious Revolution of 1688, which is why I won't claim to know the exact details on that conflict.

William and Mary ruled jointly.

Mary's father, James II converted to Catholicism and produced a Catholic heir shortly after he inherited the throne. (Mary and her sister Anne were Protestant products of James' first marriage). Fiercely Protestant Britain wasn't a fan of any of James' new pro-Catholic outlook.

William was asked to invade by some British noblemen, and James II fled England when William and Mary arrived. This was interpreted as his abdication, and it was then passed to Mary, his eldest daughter.

William and Mary required only that they be allowed to rule jointly, an important distinction, since Mary died before William. Had the throne passed to Mary alone, William would have had no right to rule after her death.