Pater rogat quid in illa oratione difficillimum sit, et filia respondet initium sibi clarissimum videri.

Because this is an indirect question: Pater rogat = The father asks ... what is most difficult.

In Latin, the verb in an indirect question is normally put in the subjunctive, not the indicative. So:

• direct question: quid ... est? = what is ... ?
• indirect question: rogat quid ... sit = he asks what ... is

So sit is the present subjunctive of esse.

Here quid is the interrogative pronoun meaning what?

The father is asking what is the most difficult thing in that speech. Because it is a question word, Latin uses quid.

That is different from:

• quod as a relative pronoun = which / that
• quod as a conjunction = because / the fact that

So in this sentence, quid is correct because the clause is asking what?

It matches quid, which is neuter singular.

The structure is basically:

• quid ... difficillimum sit = what is the most difficult thing

Since quid is neuter singular, the predicate adjective difficillimum is also neuter singular.

So Latin is treating the idea as what thing is hardest?

Yes. Difficillimum is the superlative of difficilis.

Some adjectives in -ilis form their superlatives with -illimus rather than the more regular -issimus. So:

• difficilis = difficult
• difficillimus = most difficult / very difficult

So difficillimum here means the most difficult because it agrees with neuter singular quid.

Because in with the ablative usually means in or within a place or context.

Here oratione is ablative singular after in, and illa agrees with it:

• in illa oratione = in that speech

So the idea is what in that speech is the most difficult.

Illa is a demonstrative adjective meaning that. It agrees with oratione in case, number, and gender:

• feminine
• singular
• ablative

So:

• oratio = speech
• illa oratione = that speech
• in illa oratione = in that speech

Because after verbs of saying, thinking, answering, reporting, and so on, Latin often uses an indirect statement construction.

That construction is:

• accusative + infinitive

So after respondet, instead of saying something like that the beginning seems very clear to her with a finite verb, Latin says:

• initium ... videri

This is a standard Latin pattern.

Syntactically, initium is the accusative subject of the indirect statement after respondet.

In other words, in initium sibi clarissimum videri, initium is the thing that seems.

A useful detail: the form initium looks the same in the nominative and accusative singular, because it is a neuter second-declension noun. So the ending alone does not show the case here; the construction does.

Formally, yes: videri is the present passive infinitive of videre.

But very often in Latin, videri means to seem. That is a very common idiomatic use.

So:

• initium clarissimum videri = the beginning to seem very clear

And because it is inside an indirect statement, the whole phrase means something like that the beginning seems very clear.

Sibi is the dative singular of the reflexive pronoun se. Here it means to herself / to her.

With videri, the dative often marks the person to whom something seems:

• mihi videtur = it seems to me
• tibi videtur = it seems to you
• sibi videtur = it seems to him/her/themself

In this sentence, sibi refers back to filia, the subject of respondet, not to pater.

So the sense is: the daughter replies that the beginning seems very clear to her.

Because Latin normally uses the reflexive pronoun in this kind of situation when the pronoun refers back to the subject of the main clause.

The subject of respondet is filia, so inside her reported statement Latin uses sibi to refer back to filia herself.

If Latin used ei, that would more naturally refer to someone else, not reflexively back to the daughter.

Grammatically, clarissimum is the superlative of clarus:

• clarus = clear
• clarior = clearer
• clarissimus = clearest / very clear

In many contexts, a Latin superlative can be translated either as a true superlative (clearest) or as an intensive (very clear). Which English wording sounds best depends on context.

So clarissimum videri can mean:

• to seem clearest
• or more naturally in many contexts, to seem very clear

Because Latin word order is much more flexible than English word order, and it is very common for verbs to come near the end of their clauses.

So:

• quid in illa oratione difficillimum sit
• initium sibi clarissimum videri

both have a very natural Latin feel.

Latin uses endings, not word order alone, to show how words relate to one another. That gives it more freedom to arrange words for emphasis or style.

Not as a separate word. In Latin, indirect statement usually does not need a conjunction meaning English that.

English says:

• she replies that the beginning seems very clear

Latin instead uses the accusative-and-infinitive construction:

• respondet initium sibi clarissimum videri

So the idea of English that is built into the construction itself, not expressed by a separate word.