By looking at the title, The Burial of the Dead, it is hard for the reader to make conjectures and expectations on what the poem could be about. As a matter of fact the words "burial" and "dead" suggest the idea of a funeral cerimony but the intelligent reader knows that he is in front of a modern text and it cannot be so obvious.
The part of the poem that goes from line 1 to line 23 can be divided into scenes that are not logically connected to each other.

The first scene is set in April. April is conventionally  connected to Spring, when the nature reborns, flowers blossom, the sun shines and the weather is not so cold anymore. In addition, according to the religious code, April means rebirth, resurretion of Christ.  That is why, by looking at the first line, the reader is immediately shocked and disoriented: "April is the cruellest month" the speaking voice asserts. The semanthic field build upon April with the words "dead", "Lilacs", "roots", "rain" conveys a desolated atmosphere and a totally different idea to what the common reader has been used to. Indeed the poets' poet,  Jeffrey Chaucher in The Canterbury Tales writes: " April is the sweetest month". It follows that the reader needs to ask himself: "Why is April the cruellest month?"
The second scene refers to Winter that "kept us warm, covering Earth in forgetful snow". The snow reminds to the one that covers the Dubliners in The Dead by James Joyce. It is "forgetful" because it covers and hides everything and protects those people who want to spend all their life behind a mask. The idea of this miserable life is also suggested by the adjective "little" and "forgetful" and by the word "tubers" that remarks the concept of something/somebody that/who grows underground.
The third scene opens up in Summer. As well as Spring, it is not presented in a conventional way: it is rainy. "Summer surprised us with a shower of rain"
At this point the reader finds himself in front of a German line that seems to cross the memories of the speaking voice but at the same time it underlines how people can be picky and meticulous for nothing: "I'm not Russian. I'm from Lithuania, perfect German".
As the poem goes on memories starts invading the speaking voice's mind. She remembers when her cousin took her to the mountains. The juxtaposition of images April, Winter, Summer, Austria without a chronological order  and the final position of the mountains "where you feel free" conveys the idea of the liberty and tranquillity after the tormented and desolated experience of the "cruel April".
But it is just an ullision because in the last lines the desolated and deserted landscape comes back through the use of the words "broken", "dead", "dry" and the expressions "no relief" and "no water". Water gives life and the image of a desolated land without water gives no escape.
It follows that, at the end of this first few lines of The Burial of the Dead, the reader may partially answer his initial question: "Why is April the cruellest month?". It is the cruellest because in front of desolation, desert, in front of people who only care about insignificant details, in front of a world that doesn't know anymore what the important values are, April gives the chance to nature to re-born and keep living.