Translation:She admitted that she had been longing for home.
Maybe it will help:
Esperanto tense is relative. This differs from English absolute tense, where the tense is past, present, or future of the moment of speaking: In Esperanto, the tense of a subordinate verb is instead anterior or posterior to the time of the main verb. For example, "John said that he would go" is in Esperanto Johano diris, ke li iros (lit., "John said that he will go"); this does not mean that he will go at some point in the future from now (as "John said that he will go" means in English), but that at the time he said this, his going was still in the future.
To help with this, the generaly rule states that the verb's tense in indirect speech be the same as in direct speech. So to follow mihxal's example: Johano diris, ke li iros. comes from Johano diris: “mi iros”.
Just another example:
I said that I was tired = Mi diris, ke mi estas laca
I said: “I am tired” = Mi diris: “mi estas laca”
These sentences all mean the same thing (i.e., the I-person's being tired happens in the same moment in time).