Estis necese, ke la dentisto eltiru la muelodonton.

Esperanto often uses an impersonal expression where English would use a dummy subject like it.

So English says:

• It was necessary that...

But Esperanto can simply say:

• Estis necese, ke...

There is no need for a word like English it.

Also, necese is used here because the idea is general and impersonal: it was necessary / it was necessary that... If there were a real noun being described, you would normally use necesa instead:

• La operacio estis necesa. = The operation was necessary.

But in your sentence, there is no real subject noun, so the impersonal pattern estis necese is very natural.

The ending -e makes it an adverb, and Esperanto often uses adverbs in expressions like this:

• Estas grave, ke... = It is important that...
• Estis strange, ke... = It was strange that...
• Estis necese, ke... = It was necessary that...

This is a very common Esperanto pattern.

If you used necesa, that would usually describe a noun:

• La eltiro estis necesa. = The extraction was necessary.

So in your sentence, necese does not describe a noun; it helps form an impersonal statement.

Ke introduces a subordinate clause and usually means that.

So:

• Estis necese, ke la dentisto eltiru la muelodonton.

literally works like:

• It was necessary that the dentist pull out the molar.

The part after ke is the content of what was necessary.

This is very common in Esperanto after expressions of:

• necessity
• desire
• emotion
• opinion

For example:

• Mi volas, ke vi venu. = I want you to come.
• Estas grave, ke li sciu tion. = It is important that he know that.

In Esperanto, subordinate clauses are very often separated by commas, especially when introduced by words like:

• ke = that
• ĉar = because
• se = if
• kiam = when

So the comma in:

• Estis necese, ke...

is normal and standard.

English is less regular about this, but Esperanto punctuation tends to mark clause boundaries more clearly.

The -u ending is used for the volitive/jussive mood in Esperanto. It often appears when someone expresses:

• a command
• a wish
• a request
• a necessity
• a recommendation

After estis necese, ke..., Esperanto normally uses -u because the action is something required, not just a plain fact.

So:

• ke la dentisto eltiru la muelodonton

means something like:

• that the dentist should pull out the molar
• that the dentist pull out the molar

It is not just saying what the dentist did; it is saying what needed to happen.

Because the sentence is about necessity, not a simple statement of fact.

• eltiras = pulls out / is pulling out
• eltiris = pulled out
• eltiri = to pull out
• eltiru = should pull out / is to pull out

After ke following an expression like estis necese, Esperanto usually wants the -u form.

Compare:

• La dentisto eltiris la muelodonton. = The dentist pulled out the molar.
  This is a factual statement.

• Estis necese, ke la dentisto eltiru la muelodonton. = It was necessary that the dentist pull out the molar.
  This expresses obligation/necessity.

By itself, the -u form does not primarily show tense. It shows mood: requirement, intention, wish, command, and so on.

In this sentence, the main verb is past:

• Estis necese = it was necessary

So the meaning is that, at that past time, it was necessary for the dentist to perform the extraction. The pulling out is understood relative to that past situation.

In other words, -u here is less about exact time and more about what was required.

The basic verb is tiri = to pull.

The prefix el- often means out or out from inside.

So:

• tiri = pull
• eltiri = pull out, extract

That is why eltiru is a very natural verb for removing a tooth.

The -n is the accusative ending, which usually marks the direct object.

Here, the dentist is doing the action of pulling out, and the thing being pulled out is the molar:

• la muelodonto = the molar
• la muelodonton = the molar as direct object

So:

• la dentisto eltiru la muelodonton

means:

• the dentist should pull out the molar

Without -n, muelodonto would be the subject form, not the object.

La means the, and it usually marks something definite or identifiable.

• la dentisto = the dentist
• la muelodonton = the molar

In this sentence, the speaker has a specific dentist and a specific tooth in mind, so la is natural.

If the meaning were more indefinite, you could leave out la:

• Dentisto eltiru muelodonton would sound much less natural here and much less specific.

So la helps show that we are talking about a particular dentist and a particular molar in the situation.

It is a compound word:

• mueli = to grind
• dento = tooth

So muelodonto is literally a grinding tooth, i.e. a molar.

This is very typical Esperanto word-building. Many medical or everyday terms are built from clear parts, which can make them easier to learn once you know the roots.