Long story short, Bismarck was very good at playing countries off against each other.

In a more long winded version, in 1870 the throne of Spain was sitting vacant due to a revolution. The leaders of the revolution were looking to recruit a new king form one of the noble families of Europe and ultimately offered it to Prince Leopold of Hohenzollern-Sigmaringen. Leopold was a member of a cadet branch of the the Hohenzollern family King Wilhelm of Prussia was part of, making the two distant cousins. Neither Leopold nor Wilhelm were that bothered and Leopold ultimately rejected the throne, but Bismarck saw an opportunity.

At the time, Bismarck had recently brought the North German states into a Confederation led by Prussia and was angling to get the southern states to join as well. He had already secured mutual defence pacts with the larger southern states by circulating information about Napoleon III's attempts to bargain for territory along the Rhine. What he needed now was to provoke a French attack and solidify the alliance.

After Leopold rejected the Spanish throne, word reached Napoleon III in Paris. It's important to note that Napoleon's position was looking increasingly shaky, he'd overthrown the Second Republic but was facing increasingly urgent demands for democratic reform and there were murmurs of revolution. At the same time, Prussia's star was in the ascendant. Germany had gone from a collection of small, unimportant states to an industrial powerhouse almost overnight and France's traditional position as the most powerful force in mainland Europe was under threat. The idea that Prussia and Spain could ally against France was very disturbing indeed.

This resulted in the French ambassador to Prussia, Count Vincent Benedetti, being sent to present a demand to Wilhelm II. This demand stated that the King should guarantee that a Hohenzollern prince would never be permitted to sit on the Spanish throne. The two met informally as the King was walking in the park at Bad Ems, a spa resort. Using a friendly manner, with "a courtesy that never failed him" the King refused to bind himself to any agreement and the two "departed coolly". An account of the meeting was written up and passed onto Bismarck. Wilhelm described the ambassador as having been "very importunate" and gave Bismarck permission to publish an account. This is where it gets interesting.

Bismarck changed the wording of the meeting. He removed the conciliatory phrases and made the King appear more stern. This wasn't strictly a forgery, merely emphasising the main point: France had made demands under threat of war and Wilhelm had refused. But it was designed to make Benedetti look demanding and Wilhelm look abrupt. From the German point of view, the French had insulted their king, from the French point of view, their ambassador had been insulted. Bismarck said to his friends that the edited dispatch would be like "a red rag to the Gallic bull".

After the meeting the King decided that he would no longer receive Count Benedetti and informed him through an adjutant that he had received word from Prince Leopold and had nothing further to say on the matter. This caused further problems as in German 'adjutant' refers to a relatively high ranking aide-de-camp, in French it means a non-commissioned officer. In the French view, Wilhelm had insulted their ambassador and then sent a low ranking soldier to tell him he wasn't needed any more. To make matters worse, the news broke in France on Bastille Day, before the ambassador could tell his side of the story.

The French considered their position to be much stronger than it was relative to Prussia and started to mobilise their army. Further mistranslations and interpretations only incensed the people further and on 19th of July France declared war on Prussia.

The background issue that led to the crisis was that a Hohenzollern (i.e. the ruling dynasty of Prussia) prince wanted to be elected to the spanish throne. Understandably France wasn't happy with that. It would have meant an encirclement that would be strategically unbearable. So they issued an ultimatum threatening war. This got the prussian king involved and the mounting pressure made the Hohenzollern prince's father declare that his son would bow out.

The french then send an envoy to the prussian king to ensure that the matter was settled. They wanted a promise that no Hohenzollern would try the same again. But the king had not yet recieved the news himself, so he explained his stance and asked the envoy to wait for a final answer. When the news arrived, the king had a high-ranking official (the Prussians called that an "Adjutant") tell the envoy that he simply couldn't make any such promises for the future for his entire family, as he saw it as the prince's business and not his own.

The whole story was then send to Bismark in a telegram, which became known as the Ems Dispatch. Now, the subject of meeting between the king and the envoy and how it all played out wasn't all that unreasonable. In fact the king had made quite a bit of effort to explain himself, which he wouldn't have had to and only did for diplomacy's sake. The matter was settled more or less amicably. But Bismark decided to abridge the dispatch and made it seem like the king had not had a meeting with the envoy and had not been open to understanding the french fears. Instead he made it seem like the king had been rude to the envoy. This supposed gross disrespect for France was released to the press immediately. The french news agency that translated the press release then made another error: instead of translating the differing german use of "Adjutant" they directly translated it to the french "adjutant", which meant a rather low-ranking officer. This apparent disregard for their very sensible concerns and disrespect to their envoy made the french people very angry. It was a grave insult to France.

Now, this wasn't the only reason for war. Everyone could see what was happening in Germany at the time, Prussia had already managed to unite most of the country under their banner and defeated their historic counter-balance Austria-Hungary. The French were worried about Prussia uniting Germany under a new empire, which could have threatened France itself and definitely would have(/eventually did) threatened French interests along the Rhine. They also massively over-estimated their chances of beating Prussia. Lashing out to settle the matter with arms became ever more preferable to being slowly out-grown by their upstart neighbor. The Ems Dispatch only was the final thing that pushed France over the edge to declare war.