Why can only those younger than 40 years old win the Fields Medal?

Fields wanted the award to encourage recipients' further achievement and stimulate others' renewed effort.

I am guessing that the age-restriction is put there to award "young" mathematicians. In most awards most people who receive it are pretty old. I think that until recently the Fields Medal was the most prestigious math award, now it is the Abel prize. If you look at the people who won the Abel prize, they are all "old" mathematicians. Even though there is no age restriction on the Abel prize, it seems you have to be minimum 70 years old to get it (perhaps, because it is awarded for lifetime achievement). Many people who win the Nobel prize are "old" also. Thus, perhaps Fields wanted his award prize to specifically target the young community so they get a piece of the action.

Here's the short version: When the award was proposed, those involved were specifically worried about lingering rivalry between mathematicians from opposite sides of World War I (especially French v German), a rivalry which at that point had basically sunk the first attempt at an International Mathematical Union. Fields's proposal (adopted after his death) tried to sidestep the problem of "invidious comparisons" by inserting fuzzy language about future achievement (which is hard to measure and debate) as vs just past accomplishments (which can be debated more easily and thus lead to conflict). Early Fields Medal committees interpreted this as saying that the Fields Medal should be something like an early/mid-career award for someone not already decorated with other prizes and career milestones. A big motive for doing this was to avoid getting into debates over who was really the top mathematician, especially since they had so many top mathematicians (including many they knew personally and who might be offended) to choose from. Before 1966, when the limit was codified, there were people under 40 who were ruled out as being too advanced in their careers, and also people over 40 who were seriously considered even past the stage of initial nomination.

Working on an article that explains the bigger story, but if you're around Cambridge, MA on March 6, 2017, you can see me give a preview at MIT: http://events.mit.edu/event.html?id=16503559

Also important is that the Fields Medal wasn't really in the same conversation as the Nobel Prize until 1966, and at the beginning few would have taken the comparison seriously (seeing it as more of the early-mid-career award that it was). See my article: http://www.ams.org/notices/201501/rnoti-p15.pdf