Why did a young George Washington sign a document admitting to assassinating a French military officer?

Short answer

George Washington relied on the translation of a mercenary he knew well and who had previously acted as his translator, Jacob Van Braam, and did not think he was signing a document in which (the French later claimed) he admitted assassinating a French military officer. Further, the claim that the officer killed had been on a diplomatic mission, was not mentioned in the document he signed.

Washington was actually more concerned with the part of the document which required that he and his men had to leave their arms behind and march though hostile territory; he requested that they be allowed to keep their guns to defend themselves against hostile Indians, and this request was granted. He then signed the document (copy here in French and English) and surrendered.

Details

Concerning the mistranslation and Washington's belief that he was admitting to killing a French officer (i.e. not a diplomat):

the French commander offered Washington written articles of capitulation. He read them aloud to Van Braam so there would be no question what the blurred, water-blotched handwriting said. The preamble stated that the French intended "only to avenge" the death of Jumonville. Van Braam could not translate the next word. In the guttering candlelight of Washington's makeshift command post, it looked as if it were Passailir. Witnesses later had trouble remembering whether Van Braam translated it as "death" or "loss" or "killing." What the French later contended they had written had only one possible meaning: assassination. If Washington signed his name to such an admission, he was taking full responsibility, on behalf of Virginia and the English, for the murder of a French diplomat. Washington later said that what came next was what made him willing to sign: un de nos officiers. That was easy, "one of our officers."

Source: Willard Sterne Randall, 'George Washington: A Life' (1997)

Washington considered the document "honourable" except for one part:

What concerned Washington more at the moment was that, after guaranteeing the Virginians' safe passage, the French wanted to "reserve" to themselves all his artillery "and munitions of war." This meant that Washington and his men would have to leave behind their guns and gunpowder and walk fifty miles through forests swarming with hostile Indians whom Villiers could only promise to restrain....he refused to sign. The French commander ended the impasse with a single stroke of his pen through the offending words.

Source: Randall

Ron Chernow, in Washington: A Life, has a broadly similar account, but emphasizes Washington's unawareness of the word 'assassination' in the document.

How could this profound misunderstanding have arisen? The night of the negotiation was dark and rainy, and when Van Braam brought back the terms of surrender, Washington and the other officers strained to read the blurry words in a dim light. “We could scarcely keep the candlelight to read them,” recalled one officer [see note 1 below]. “They were wrote in a bad hand, on wet and blotted paper, so that no person could read them but Van Braam, who had heard them from the mouth of the French officer.” Not expert in English, Van Braam might have used death or loss interchangeably with assassination, yet it’s hard to imagine that he botched the translation deliberately. What is clear is that Washington was adamant that he never consented to the loaded word assassination. “That we were willfully, or ignorantly, deceived by our interpreter in regard to the word assassination, I do aver and will to my dying moment,” Washington insisted.

On this last statement by Washington, more can be found here. He disputes much of his adversary's version of events, but also makes a fairly wild claim as to how many French were killed. Washington also disputed claims by the prisoners captured from Joseph Coulon de Villiers, Sieur de Jumonville's force (claims later re-iterated by the French) that they had been on a diplomatic mission when de Jumonville was killed [see note 2].

As to why Washington signed a document at all, the terms offered by the French essentially required that the conditions be put in writing; there were wider issues at stake which commanders of both sides would need to see. Further, Washington was not in a good position: the British were running low on supplies, they were outnumbered, their gunpowder was damp, the terrain favoured the French due to the poor location of the fort, and the fort itself was too small to provide shelter from the wet weather for all the soldiers. Under the circumstances, Washington negotiated the best he could [see Note 3 below]. The French commander Louis Coulon de Villiers' final offer was:

If the English were now prepared to sign articles of capitulation, to withdraw from the Ohio Country and pledge not to return within the space of a year, to repatriate the prisoners they had taken, and to leave two officers as hostages at Fort Duquesne to guarantee the fulfillment of the surrender terms, he would allow them to march off the next day carrying their personal possessions, their arms, and their colors. But if the English did not agree to these terms, Coulon assured the Dutchman [Van Braam], he would destroy them.

Source: Fred Anderson, 'Crucible of War: The Seven Years War and the Fate of Empire in British North America, 1754-1766

Added to this, Washington was young and inexperienced, and he was relying on someone (Van Braam) who had previously served with his older half-brother. Van Braam had already acted successfully as an interpreter and Washington would have had no good reason not to trust him (Van Braam was later cleared of charges of treason in mistranslating the French terms).

As Evargalo has noted in a comment, this incident should not be overestimated with regard to the Seven Years War; there were many causes and states involved, and territorial disputes were widespread.

Notes

Note 1: The officer mentioned is Adam Stephen, a Scottish doctor and military officer who was with Washington. This National Archives footnote for the page II., 3 July 1754 has more:

In his discussion of the terms of capitulation Adam Stephen stated that when Van Braam “returned with the French Proposals, we were obliged to take the Sense of them by Word of Mouth: It rained so heavily that he could not give us a written Translation of them; we could scarcely keep the Candle light to read them; they were wrote in a bad Hand, on wet and blotted Paper, so that no Person could read them but Van Braam, who had heard them from the Mouth of the French Officer. Every Officer, then present, is willing to declare, that there was no such Word as Assassination mentioned; the Terms expressed to us, were ‘the Death of Jumonville.’ If it had been mentioned, we could have got it altered, as the French seemed very condescending, and willing to bring Things to a Conclusion, during the whole Course of the Interview” (Maryland Gazette [Annapolis], 29 Aug. 1754).

Note 2: The following is from Washington's Expedition to the Ohio, 1754: Narrative:

They [the French prisoners] informed me that they had been sent with a Summons to order me to depart. A plausible Pretence to discover our Camp, and to obtain the Knowledge of our Forces and our Situation! It was so clear that they were come to reconnoitre what we were, that I admired at their Assurance, when they told me they were come as an Embassy; for their Instructions mentioned that they should get what Knowledge they could of the Roads, Rivers, and of all the Country as far as Potowmack: And instead of coming as an Embassador, publickly, and in an open Manner, they came secretly,...

Note 3: The National archives has this footnote:

As Adam Stephen pointed out, GW had little option but to surrender, and the fact that the French were willing to offer terms was “no disagreeable News to us, who had received no Intelligence of the Approach of our Convoys or Reinforcements, and who had only a Couple of Bags of Flour and a little Bacon left for the Support of 300 Men.” Most of the arms were out of order, and the powder was wet from exposure to the incessant rain. “But what was still worse, it was no sooner dark, than one half of our Men got drunk” (Maryland Gazette, 29 Aug. 1754). The men had broken into the fort’s liquor supply. According to GW’s later testimony on the capitulation of the fort to the French, he “absolutely refused their first and second proposals and would consent to capitulate on no other terms than such as we obtained.”