The poem begins by describing a series of violent political crises, all of which, according to the speaker, trace back to a single event. Someone's "hand," by the act of "sign[ing]" a single "paper," has caused a domino-like series of catastrophes. The speaker describes this process in compressed, indirect, metaphorical language:
The hand that signed the paper felled a city;
Five sovereign fingers taxed the breath,
Doubled the globe of dead and halved a country;
Since the hand and its fingers are "sovereign" (self-ruling, like a monarch or autonomous country), the "paper" would seem to be an official document signed by a world leader. Later lines will reveal that the paper is, in fact, a diplomatic "treaty"—one designed to end a war or other "murder[ous]" conflict. But from the start, it's obvious that the treaty's impact has been anything but peaceful. It has started a political ripple effect that destroyed a city ("felled" it like a tree), ripped an entire "country" in "hal[f]" (forcibly split it into two separate nations), and apparently, killed millions of people. The treaty—or, rather, the leader who signed it—has "Doubled the globe of dead," or doubled the number of corpses in the world. The leader has metaphorically "taxed the breath" of these victims, as if confiscating their very lives through some cold bureaucratic process.
Even if the speaker is being hyperbolic, it's clear that they're accusing this leader of something like genocide. Whatever the leader intended to achieve by signing the paper, they have actually caused mass death. They've also caused the death of a fellow world leader, who is either a literal or metaphorical "king." As the speaker dryly puts it: "These five kings did a king to death." In other words, the treaty-signing leader wielded more power in each finger of their hand than the leader who got assassinated!
Notice that the "hand"/"fingers" here function as a synecdoche for the leader as a whole, who is never identified or described in detail. The "hand" seems almost detached from the rest of the leader, as though it were a mere extension of the government (or of the long arm of the law). Over the course of the poem, the "hand" becomes a symbol for political power itself.
If the historical references here seem vague, that's because they are. Thomas doesn't tie the poem to a particular treaty, leader, country, or era. This ambiguity makes the poem's political commentary more universally applicable. Thomas was writing during a period (the 1930s) that was rife with dictators, wars, and controversial treaties, so he might be alluding to events in his own time—or he might not. After all, every age of human history has seen leaders who claim more than their share of power and cause vast suffering as a result.