Mater dicit scopam novam emendam esse, quia vetus scopa nimis brevis est.

Because Latin usually expresses indirect statement after verbs like dicit with an accusative + infinitive construction.

So instead of using a separate word for that, Latin does this:

• scopam = the subject of the reported statement, put into the accusative
• emendam esse = the infinitive part

A useful comparison is:

• Direct statement: Scopa nova emenda est. = A new broom must be bought.
• Indirect statement: Mater dicit scopam novam emendam esse. = Mother says that a new broom must be bought.

Yes, it is the logical subject of emendam esse, but in Latin indirect statement, the subject of the infinitive goes into the accusative.

So in English we think:

• a new broom must be bought

But in Latin after dicit, the subject becomes accusative:

• scopam novam emendam esse

This is one of the most important Latin patterns to get used to:

• verb of saying/thinking/perceiving
• then accusative subject + infinitive

Emendam esse means to have to be bought or more naturally must be bought.

It is made from:

• emendam = the gerundive of emo (buy)
• esse = to be

The gerundive + esse forms the passive periphrastic, which usually expresses necessity or obligation.

So:

• emenda est = must be bought
• emendam esse = to have to be bought / must be bought inside indirect statement

Very literally, it is something like is needing to be bought, but English usually translates it more simply as must be bought.

Because emendam agrees with scopam.

The gerundive behaves like an adjective, so it must match the noun it goes with in:

• gender
• number
• case

Here:

• scopam = feminine singular accusative
• so the gerundive must also be feminine singular accusative
• therefore: emendam

If the noun were nominative, you would get emenda instead:

• Scopa nova emenda est.

But after dicit, the subject of the indirect statement is accusative, so the form changes to emendam.

Because emendam by itself is only an adjective-like form. To make the full idea must be bought, Latin needs the verb esse.

So:

• emendam = needing to be bought
• emendam esse = to be needing to be bought = must be bought

And because this whole part is inside indirect statement, Latin uses the infinitive esse, not a finite verb like est.

Compare:

• Direct: Scopa nova emenda est.
• Indirect: Mater dicit scopam novam emendam esse.

Because the two broom phrases have different grammatical jobs.

In the first part:

• scopam novam emendam esse
• this is indirect statement after dicit
• so scopam is accusative

In the second part:

• quia vetus scopa nimis brevis est
• this is an ordinary clause introduced by quia
• vetus scopa is the subject
• so it is nominative

So the difference is not about new versus old. It is about case:

• scopam novam = accusative
• vetus scopa = nominative

Because vetus belongs to a different adjective pattern from novus, nova, novum.

• novam comes from novus, nova, novum, a 1st/2nd-declension adjective
• vetus is a 3rd-declension adjective

So they do not decline in the same way.

Here, vetus is nominative feminine singular, agreeing with scopa. Its form happens to be vetus.

If it were accusative feminine singular, you would see:

• veterem scopam

So the sentence has:

• vetus scopa = the old broom as subject
• not veterem scopam, because this clause needs nominative, not accusative

Because brevis agrees with scopa, which is feminine singular.

• scopa = feminine singular
• therefore the adjective must be feminine singular too
• so: brevis

For this adjective:

• masculine/feminine nominative singular = brevis
• neuter nominative singular = breve

So:

• scopa brevis = a short broom
• templum breve would be wrong, because neuter needs breve

Nimis is an adverb meaning too, excessively, or overly.

It modifies brevis:

• brevis = short
• nimis brevis = too short

So vetus scopa nimis brevis est means the old broom is too short.

This is the reason introduced by quia.

Because quia introduces a regular subordinate clause of reason: because the old broom is too short.

So Latin simply uses an ordinary clause:

• quia vetus scopa nimis brevis est

That clause has:

• subject: vetus scopa
• predicate adjective: nimis brevis
• verb: est

The indirect statement belongs to dicit:

• scopam novam emendam esse

Then quia adds the reason.

The word order is fairly flexible.

Latin relies much more on endings than English does, so many rearrangements are possible without changing the basic meaning. For example, you could also have:

• Mater dicit novam scopam emendam esse
• quia scopa vetus nimis brevis est

The current order is natural and clear, but not the only possible one.

A rough way to think about it is:

• endings show grammatical function
• word order often shows emphasis, style, or what the speaker wants to foreground

Yes. With this kind of construction, Latin can add a dative of agent to show the person on whom the obligation falls.

For example:

• Mater dicit mihi scopam novam emendam esse.

That would mean:

• Mother says that I have to buy a new broom
• or literally that a new broom is to be bought by me

In your sentence, no person is stated, so the idea is just a new broom must be bought, without saying exactly by whom.