To describe the meeting as "provoking" WWI is sensationalist and incorrect. This Telegraph article quotes the letter more extensively, and makes clear that it wasn't the kings words in a letter but Grey's nephews reporting of the Kings words based on his recollection of events reported to him in 1933, twenty years after the fact. It's interesting, but seems to me to be side note for the following reasons:

The war had already broken out by the time of the meeting on August 2nd. Austria-Hungary had declared war on Serbia on 28 July. This prompted Russian mobilization. Once Russian mobilization began there was still some question (albeit an unrealistic one) of France staying out of the war, negotiated by Grey. But the fact of Russian mobilization is what set German plans for a general European war in motion.

On July 31 Germany issues an ultimatum to Russia to halt its mobilization, Russia refuses. Germany declares war on Russia on August 1. On August 2nd, the same day of this meeting, Germany issues its ultimatum to Belgium and invades Luxembourg. On 3 August, Germany declares war on France. On 4 August, Germany invades Belgium.

It's the fact of the German invasion of Belgium that forces the British cabinet's decision to join with France and Russia and to send a British expeditionary force to France. On August 4th, Britain issued an ultimatum to Germany to respect Belgian neutrality. That ultimatum expired at 11 o'clock on August 4th and Britain declared war on Germany.

None of that was "provoked" by a meeting between the king and the foreign secretary on 2 August.

I hadn't known about this letter, but frankly it's not that shocking. Grey was already arguing to cabinet that Britain had a moral duty to side with France in the war, not least on the basis of the 1912 naval agreement but also based on secret negotiations he had initiated with France in 1906. His pro-French anti-German stance was not universally held by the cabinet, though was apparently shared by the king, which is interesting, but of limited relevance in the British system of government (in strong contrast with some of the other great powers which were more autocratic.)

As to the Rothschilds, I'm not sure where you're reading that into this.