La luota dometo estas proksime de la strando, kaj mia fratino diras, ke ŝi volas lui ĝin por unu semajno.

Luota comes from lui + -ot- + -a.

• lui = to rent
• -ot- = a passive participle meaning to be ...ed or meant to be ...ed
• -a = adjective ending

So luota dometo means something like a cottage to be rented or, more naturally in English, a cottage for rent.

The ending -et- is a diminutive suffix. It adds the idea of smallness, and sometimes also a slightly warm or affectionate tone.

So:

• domo = house
• dometo = small house, little house, cottage

That is why dometo is a good match for cottage.

Here Esperanto uses the expression esti proksime de, which means to be near or to be close to.

So estas proksime de la strando means is near the beach.

You could also use the adjective form, but then the structure changes:

• La dometo estas proksime de la strando
• La dometo estas proksima al la strando

Both are natural, but the sentence you have uses the adverbial pattern.

With proksime, Esperanto commonly uses de:

• proksime de la strando = near the beach

If you use the adjective proksima, then al is common:

• proksima al la strando

So the choice of preposition goes together with the form being used.

Ke introduces a subordinate clause. It is the Esperanto word for English that in a sentence like my sister says that she wants...

So:

• mia fratino diras, ke ŝi volas lui ĝin
  = my sister says that she wants to rent it

In English, that is often omitted, but in Esperanto ke is usually kept.

Because Esperanto, like English, often uses a pronoun instead of repeating a noun.

• ĝi = it
• ĝin = it as a direct object

Since the cottage has already been mentioned, ĝin is the natural way to refer back to it.

The -n marks the direct object.

In this part of the sentence:

• ŝi = the person doing the action
• volas lui = wants to rent
• ĝin = the thing being rented

Esperanto adds -n to direct objects, including pronouns. So ĝi becomes ĝin here.

By itself, lui can be a little broad, and context usually tells you which direction is meant.

In this sentence:

• ŝi volas lui ĝin clearly means she wants to rent it
• luota dometo means a cottage meant to be rented or for rent

So even though English separates rent and rent out, Esperanto often lets the context do that work.

Por here means for in the sense of for a period of.

So:

• lui ĝin por unu semajno = rent it for one week

Because por already shows the relationship, semajno does not need an -n.

If there were no preposition, Esperanto could show duration with -n, as in unu semajnon. But with renting, booking, or reserving something, por unu semajno is very natural.