Mia frato volas ordigi la tablon antaŭ la kunveno.

Because mia is the possessive form of mi.

• mi = I
• mia = my

In Esperanto, possessive words behave like adjectives, so they usually end in -a:

• mia frato = my brother
• via tablo = your table
• ŝia libro = her book

So mi frato would be ungrammatical.

Because mia already makes the noun definite enough in normal Esperanto.

• mia frato = my brother
• not usually la mia frato

Using la together with a possessive is usually unnecessary. English works similarly: we say my brother, not the my brother.

Because Esperanto often uses a conjugated verb plus an infinitive, just like English does in sentences such as wants to tidy.

Here:

• volas = wants
• ordigi = to put in order / to tidy up

So the structure is:

• Mia frato volas ordigi...
• My brother wants to tidy...

Only volas is conjugated for tense. ordigi stays in the infinitive form.

In Esperanto, the infinitive ending -i already carries the idea of English to before a verb.

So:

• ordigi = to tidy / to arrange
• vidi = to see
• manĝi = to eat

That means Esperanto does not need an extra word here.
volas ordigi literally works like wants tidy-to-do, but in natural English we translate it as wants to tidy.

The suffix -ig- usually means to cause to become or to make something do/become something.

Here the base idea is ordo = order.
So:

• ordo = order
• orda = orderly
• ordigi = to make orderly, to put in order, to tidy up

This is a very useful Esperanto pattern:

• pura = clean

• purigi = to clean, to make clean

• larĝa = wide
• larĝigi = to widen, to make wider

So ordigi la tablon means to make the table orderly.

Because la tablon is the direct object of the verb.

In Esperanto, the direct object usually takes the accusative ending -n.

Here, the brother is doing the action of ordigi to the table, so:

• la tablo = the table
• la tablon = the table as the direct object

Compare:

• La tablo estas granda. = The table is big.
  No -n, because it is the subject.

• Mi vidas la tablon. = I see the table.
  -n, because it is the object.

Because only tablon is the direct object.

• la tablon is what gets tidied, so it takes -n
• antaŭ la kunveno is a prepositional phrase, and kunveno is the object of the preposition antaŭ

Normally, prepositions in Esperanto do not require -n on the following noun. So:

• antaŭ la kunveno = before the meeting

The -n is not there because kunveno is not the direct object of the main verb.

Here antaŭ means before in a time sense.

So:

• antaŭ la kunveno = before the meeting

But antaŭ can also mean in front of in a physical sense:

• La seĝo estas antaŭ la tablo. = The chair is in front of the table.

So antaŭ can be either:

• temporal: before
• spatial: in front of

In this sentence, it is clearly temporal.

Because it refers to a specific meeting, not just any meeting.

Esperanto has:

• la = the
• no separate word for a/an

So:

• la kunveno = the meeting
• kunveno by itself can mean meeting in a more general sense

The sentence suggests a particular meeting that the speaker and listener can identify.

Not completely. Esperanto word order is fairly flexible, especially because the -n ending helps show what is the object.

The neutral order here is:

• Mia frato volas ordigi la tablon antaŭ la kunveno.

But you could also say:

• Antaŭ la kunveno mia frato volas ordigi la tablon.

That puts more emphasis on before the meeting.

Still, the original version is the most straightforward and natural for a learner to use.

It can sometimes overlap in meaning, but it is not exactly the same as clean the table.

• ordigi = put in order, tidy up, arrange
• purigi = clean, make clean

So if the idea is:

• remove clutter, organize papers, make the table neat -> ordigi
• wipe dirt off the surface -> purigi

In real life, tidying and cleaning often happen together, but Esperanto keeps the core meanings distinct.

A lot of Esperanto grammar is shown by endings. In this sentence:

• mia: -a adjective/possessive form
• frato: -o noun
• volas: -as present tense verb
• ordigi: -i infinitive verb
• tablon: -o noun + -n direct object
• kunveno: -o noun

This is one of the biggest advantages of Esperanto: once you know the endings, sentences become much easier to decode.