A poem is basically something beyond common understanding. As Will pointed out, it speaks for everybody and in a different way.

When Royokan, Ikkyu, Basho, and so on write, they are writing to you. To you personally. They are writing in our place; about what we have, but do not yet understand.
Or the very beautiful answer of Tobiishi:

I read a different translation of that poem a while back, and wrote this one in response. I have difficulty commenting on poetry with anything other than more poetry-

So what I am about to undertake is neither easy nor definitive. Just a few clues really. A few pointers.
A bunch of modest and simple twigs, a bit of dry wood. Nothing fancy. No Teisho. No kusen. No Japanese exotic whatever.

Before we even start to speak, before we are about to read, there is a moment. In this moment words are not yet manifested, things not yet trapped in sounds. Or so it seems.

I've built a grass hut where there?s nothing of value.
After eating, I relax and enjoy a nap.

We don't need to rush into the poem, we can linger and take our time in this blank space of the page, the unwritten part of the screen, the silent body-mind of ours. Because all the words we are about to read leap clear from there, they spring and come from this undefined field, this original openess and clarity of THIS and return to it. What I am just suggesting now is that a poem starts long before its first words are uttered and lasts a longtime before its final verse. This basic scenery is what our ancestor Keizan Jokin (1268-1325) is pointing at at the beginning of his Zazen Yojinki:

This is called "the display of the Original Face" and "revealing the landscape of the basic ground".

This is what Zazen is, the display of the original face. But saying this is already turning one into two. Saying these words are extra, breaking the beautiful silence, somehow imperfect in themselves is another illusion. The metaphors of Dogen, Keizan, and many Chinese ancestors are constantly pointing at the source, the origin ( think about the Sandokai :The spiritual source shines clear in the light;the branching streams flow on in the dark of Sekito Kisen). Why is that? Did we loose anything? In fact, the poem is not interrupting this, words are like footsteps in the snow, snow-made too. You cannot dream of speaking the words and discriminating without that ever present source. It is always there. And in fact it is just you. Nothing and nobody but you. Your body here and not different from Buddha' s body, truly non-dual, which is neither one, nor two. Clarity is not something that happened before or you will get later as a prize or nice spiritual candy for all our good work, it the very fabric of your body-mind. Sitting is allowing this to unfold freely, to live freely, to see within the impermanence itself the most direct clarity.It strikes me that most people think that music is only about making sounds, but in many ways, music is about articulating silence, and if you pay attention silence is loud and sounds very quiet!

Let me put it another way, we all know the famous story about the finger pointing at the moon. The traditionnal understanding of this is that we musn't fool ourselves and identify finger and moon. In our Soto tradition, the moon is seen within the finger and the finger pointing the moon is the moon too!
Sometimes the moon points at the finger, sometimes the moon points at the moon. Sometimes, it is just just one big bright moon-pearl. It permeates the four corners of the kesa-world, brightness, just brightness ( which is beyond bright and dark) Anyway...

Once we really breathe and bring yourself down to this, we may read the fist verse.

What a wonderful thing to see that we don't sit to become Buddha, but rather, because we are Buddha, we sit. So you may read this poem under the hut roof. What is this hut? Well, in the context, it refers of course to the life of simple ermits and sages of the past, seeking simplicity in the mountain. This hut may represent an easy, very natural life-style where everything is simple again. But if am sure you already understand that this all poem is also a metaphor of shikantaza itself. The grass hut is this body-mind sitting in as-it-isness.

I?ve built a grass hut where there?s nothing of value

There is nothing of value means everything is very precious. A speck of dust, a blade of grass, a piece of s...., a dead cat or anything is a treasure in the true sense. Everything is a wonder. New. Common sense of value is a byproduct of a frozen perception of time, value is a tag we put to things, stuff and people because of our past experiences or cultural beliefs. Wonder is what arises fresh when we let go of our values. Values are like the toys we think are so important. The toys and notions we constantly play with. In shikantaza we are invited to experience without keeping, controling, measuring, chasing. Our appreciation is in the reality as is, raw, unrefined. Noisy, noisy. Crowdy,crowdy. Empty, empty. So what? Just this and this arises and this fades away. Building the grass hut is sitting the body-mind in this constant change. This hut is always available, always being built. In a way, I kind of sense that the use of the past could be a betrayal of the original meaning, a bit like the opening sentence of the Heart Sutra: when Avolokitesvara(...) perceived. Why not perceives? Why not I am building or something like it? Nothing of value is the opening verse of the Shin Jin Mei of Sozan:

The great Way is not difficult, just avoid picking up and choosing(...)

nothing of value is our runny nose, our screaming child, our tummy ache, the dog barking, the sound of next door television or the racket of cicaddas. Nothing of value has nothing to do with cheap living, quite the opposite, the way to see through and appreciate totally, thoroughly our life (...)