Skribu la nombron de ŝlosiloj en la inventaron.

-U is the Esperanto verb ending for a command, request, wish, or instruction. So skribu means write in the sense of telling someone to do it.

A few useful points:

• skribi = to write
• skribu = write

This form does not change for singular vs. plural you. It can mean:

• write to one person
• write to several people
• even a polite instruction, depending on context

So Skribu... is a normal way to give an instruction.

The -n marks the direct object.

Here, the thing being written is la nombro de ŝlosiloj or, in this sentence, la nombron de ŝlosiloj because it is the object of skribu.

So the structure is:

• skribu = write
• la nombron = the number

In Esperanto, the direct object usually takes -n. English does not mark this with a special ending, but Esperanto does.

Esperanto uses nombro de ... for number of ....

So:

• la nombro de ŝlosiloj = the number of keys

This is the normal pattern. After de, the noun does not become the direct object of the sentence, so it does not take the object -n here.

Also, ŝlosiloj is plural because the phrase means keys as the category being counted.

Compare:

• la nombro de libroj = the number of books
• la nombro de homoj = the number of people

This is one of the most important things in the sentence.

After a preposition, Esperanto can use -n to show direction or movement toward a destination. Here, en la inventaron means something like into the inventory.

So the idea is not just location, but entry into that place or record.

• en la inventaro = in the inventory
• en la inventaron = into the inventory

In this sentence, the information is being entered into the inventory, so en la inventaron is natural.

This use of -n after a preposition is often called the directional accusative.

Because both nouns are understood as definite:

• la nombron = the specific number you are supposed to write
• la inventaron = the specific inventory being referred to

Esperanto uses la for the, and it does not change form. It stays la whether the noun is singular, plural, object, or not.

So:

• la nombro
• la nombroj
• la nombron

all keep la unchanged.

Yes, plural is the normal form in this pattern.

When you say the number of keys, you are talking about how many items there are in the category keys, so Esperanto uses the plural:

• la nombro de ŝlosiloj

Even if the actual answer might turn out to be 1, the expression still normally stays plural, because it means the number of keys, not the number of key.

Yes. Esperanto word order is fairly flexible because endings like -n show what each word is doing.

For example, these are both understandable:

• Skribu la nombron de ŝlosiloj en la inventaron.
• Skribu en la inventaron la nombron de ŝlosiloj.

The original version puts the focus first on what should be written, and then says where it should be written.

So the given word order is normal and clear.

Usually, inventaro means an inventory, that is, a list, register, or record of items, not the storage room itself.

So in this sentence, en la inventaron means entering the information into the inventory record.

That is why the sentence feels natural with a verb like skribu. You write something into the inventory as a record.