Something I always hear in WWI was that at a time around christmas, the armies had a truce where they wouldnt fight, instead, they would just greet the others and partake in friendly activities like soccer and exchange gifts, how true was this? Was it a formal truce, or just something all the soldiers did? How did somebody like the German Kaiser respond to this truce? Is it all just some rumor, or is there some truth to the story?
Yes, there was a truce. It took place in Christmas 1914 (there were also much more limited attempts at fraternisation in 1915, and on the Eastern Front there was an equivalent truce between Russians and Austro-Hungarian troops at Easter 1916.)
It was informal and local, generally initiated on the German side, and more often involved German and British troops than British and French, probably because the French, who had been invaded, felt more personal animosity to the Germans than the British did at this point in the war.
There was not one football match, but several, rather informal and chaotic ones. Gifts were certainly exchanged between men of the two sides, again on an informal and one-on-one basis, but the truce itself, and the cessation of hostilities just for a day or so, was the main "present" that the men received.
The reaction of both high commands was horror, because the truce threatened not only military discipline but potentially even their ability to prosecute the war. Strict measures were put in place in future years to prevent it happening again, with heavy punishments mandated for any who attempted to take part in a repetition. Those who participated in 1914 were not reprimanded on the whole because there had been no explicit orders forbidding it, and far too many, of all ranks, had participated for punitive action to be possible.
I wrote about the truce in much more detail, with a focus on the football matches, here. And I discussed the disciplinary aspects of the truce in an earlier answer here.