La konsulo bazas la peton sur la legitimilo.

Because peton is the direct object of the verb bazas.

In Esperanto, nouns take -n when they are the direct object of the action:

• La konsulo bazas la peton... = The consul bases the request...
• La peto would be the subject form.
• La peton is the object form.

So the consul is doing the action, and the request is what is being based on something.

It means bases ... on ... or grounds ... on ....

The pattern is:

• bazi ion sur io = to base something on something
• La konsulo bazas la peton sur la legitimilo.

So:

• la peton = the thing being based
• sur la legitimilo = the thing it is based on

This works very much like English to base something on something.

Yes, sur often literally means on or on top of, but it is also used figuratively, just like English on in based on.

So in this sentence, sur la legitimilo does not mean the request is physically sitting on the credential. It means the request is founded on or supported by the credential.

This figurative use is normal and idiomatic.

Legitimilo is a noun meaning something like credential, ID document, or proof of identity/status, depending on context.

It is built from parts:

• legitim- = a root related to legitimacy or authorization
• -il- = instrument/tool suffix
• -o = noun ending

So legitimilo is literally something like a tool/document for legitimation, which in practice means an identification or credential document.

The suffix -il- is very common in Esperanto:

• tranĉi = to cut → tranĉilo = knife
• skribi = to write → skribilo = writing instrument
• legitim- → legitimilo = credential/ID document

Because la is the Esperanto definite article, meaning the.

So:

• La konsulo = the consul
• la peton = the request
• la legitimilo = the credential / the ID document

Esperanto uses la when something is definite or identifiable in context, just like English the.

Also, Esperanto has no indefinite article. So:

• konsulo can mean a consul or just consul, depending on context
• la konsulo specifically means the consul

Yes. The ending -as marks the present tense in Esperanto.

So:

• bazas = bases / is basing
• bazis = based
• bazos = will base
• bazu = base! / let ... base
• bazus = would base

In this sentence, bazas simply tells you the action is in the present.

Yes, to some extent. Esperanto word order is fairly flexible because grammar endings show what each word is doing.

For example, peton has -n, so it is clearly the object.

Still, the given order is the most neutral and easiest to understand:

• La konsulo bazas la peton sur la legitimilo.

That is the standard subject–verb–object style. Other orders are possible, but they may sound more marked, more literary, or less natural in ordinary use.

Because la peton just means the request, without saying whose request it is.

If the meaning were specifically his/her own request, Esperanto would often use sian:

• La konsulo bazas sian peton sur la legitimilo.

Here, sian is the reflexive possessive, used when the possessor is the subject of the clause.

So the difference is:

• la peton = the request
• sian peton = his/her own request

The original sentence leaves possession unspecified.

Yes. Esperanto often uses baziĝi sur for to be based on.

So a very common alternative would be:

• La peto baziĝas sur la legitimilo.

That means:

• The request is based on the credential.

Compare:

• La konsulo bazas la peton sur la legitimilo. = The consul bases the request on the credential.
• La peto baziĝas sur la legitimilo. = The request is based on the credential.

Both are correct; they just present the idea differently.

Esperanto pronunciation is regular, with stress usually on the second-to-last syllable.

So:

• konsulo → kon-SU-lo
• bazas → BA-zas
• peton → PE-ton
• sur → sur
• legitimilo → le-gi-ti-MI-lo

A few useful pronunciation reminders:

• e is like e in get, not like English ee
• o is a pure o sound
• u is like oo in food
• every letter is pronounced clearly

That regularity is one of the nice things about Esperanto.