The Descent of Inanna (Ishtar) was unearthed from 1889 - 1900, but as far as I can tell, it wasn't really considered that well reconstructed until around the 1940s or later. Early published versions were based on less complete information, and had to make assumptions that turned out incorrect, especially with regards to Dumuzi's (Tammuz) role in the story (See: Kramer's explanation of the decipherment in this print version, and also "Tammuz and the Bible").
I will be using Wolkstein and Kramer's translation of the text, published in 1983. Both because it's excellent and well-explained, and because it's the one sitting on my desk. Full text here.
So, we could take Inanna (Ishtar) at her word. Her overt reason is given:
"Because . . . of my older sister, Ereshkigal,
Her husband, Gugulanna, the Bull of Heaven, has died.
I have come to witness the funeral rites.
Let the beer of his funeral rites be poured into the cup.
Let it be done.
That, however, is just the excuse she gives at the gate. It doesn't seem likely this is her real reason for the trip. More likely the real reason is shown here:
From the Great Above she opened her ear to the Great Below.
From the Great Above the goddess opened her ear to the Great Below.
From the Great Above Inanna opened her ear to the Great Below.
Which gives us the hint that she has heard the Great Below, the underworld, in some way. Wolkstein's explanations, however, give us more to go on:
Inanna is the Queen of Heaven and Earth, but she does not know the underworld. Until her ear opens to the Great Below, her understanding is necessarily limited. In Sumerian, the word for ear and wisdom is the same ... In order to fully appreciate or "know" what is said or meant, a great understanding is needed - an understanding of all things. It is the Great Below, and knowledge of death and rebirth, life and stasis, that will make Inanna an "Honored Counselor" and a guide to the land.
So Inanna, already powerful and knowledgeable of Heaven and Earth, needs to acquire knowledge of death and the underworld to be complete in her wisdom. This is further supported later, from the statement of Enlil, when hearing that Inanna has not returned from the underworld:
"My daughter craved the Great Above.
Inanna craved the Great Below.
She who receives the me of the underworld does not return.
She who goes to the Dark City stays there."
Here Enlil (and later, Nanna, in an identical statement) states that Inanna has gone seeking the me of the underworld. me is a tricky word that most translators refuse to translate, and too much can be said about it to address completely here. Suffice to say, the me represent knowledge and wisdom. So, the gods know that she has gone seeking that knowledge that can only be gained in the underworld.
To go just one more step, a very popular interpretation of the text is through a Jungian lens. Just to hit the main points very briefly.
The goal of gaining wisdom of the underworld, to complement the wisdom of the heavens and of the earth, speaks of Inanna becoming complete, but this interpretation views it, instead of her completing her body of knowledge, she is completing herself, developing into a whole person (goddess).
In this light, we see Inanna and her sister Ereshkigal as part of the same whole person. In order to become complete, Inanna must become vulnerable, and face the dark side of herself (Ereshkigal), die and be reborn.
Inanna abandons her temples on the way to the underworld, and has to leave her crown, robe and other vestments (also me. Like I said, complicated) at the gates of the underworld. She arrives naked, powerless and vulnerable. The death and rebirth are fairly clear to be read, but of particular interest is that after Inanna's death, and preparing for rebirth, Ereshkigal is in pain "With the cries of a woman about to give birth." Which does rather speak to the interpretation of being reborn from her darker half.