Ŝi verŝajne adoptos la hundidon.

Because -os is the Esperanto ending for the future tense.

• adopti = to adopt
• adoptas = adopts / is adopting
• adoptis = adopted
• adoptos = will adopt

One nice thing about Esperanto is that verb endings are very regular, so once you know -os, you can recognize the future tense in any verb.

Verŝajne is an adverb, and many Esperanto adverbs end in -e.

Here it means probably.
It is related to verŝajna, which means probable.

So:

• verŝajna = probable
• verŝajne = probably

This is a very common pattern in Esperanto:

• bona = good
• bone = well

The -n marks the direct object.

In this sentence, the puppy is the thing being adopted, so hundido becomes hundidon.

Compare:

• La hundido dormas. = The puppy is sleeping.
  
  • Here hundido is the subject, so no -n.
• Ŝi adoptos la hundidon. = She will adopt the puppy.
  
  • Here hundidon is the direct object, so it takes -n.

This is called the accusative ending.

• hundo = dog
• hundido = puppy / young dog

The important part is -id-, which often means offspring or young one.

So:

• kato = cat
• katido = kitten

This is different from a suffix like -et-, which means small. A hundeto would be a small dog, not necessarily a puppy.

La is the Esperanto word for the.

It does not change for gender, number, or case:

• la hundo = the dog
• la hundoj = the dogs
• la hundon = the dog (object)
• la hundojn = the dogs (object)

So in la hundidon, la stays exactly the same.

Yes. Esperanto word order is fairly flexible, especially because the -n ending helps show what the object is.

So these can all work, with slightly different emphasis:

• Ŝi verŝajne adoptos la hundidon.
• Verŝajne ŝi adoptos la hundidon.
• La hundidon ŝi verŝajne adoptos.

The original sentence is perfectly normal.
A version beginning with Verŝajne may sound especially natural if you want to emphasize probably first.

Ŝi is pronounced roughly like shee.

The letter ŝ is pronounced like English sh in ship.

So:

• ŝi = shee

Also, ŝi is the pronoun for she.

Roughly: ver-SHAI-ne

A few useful points:

• ŝ = sh
• aj sounds like English eye
• the stress in Esperanto normally falls on the second-to-last syllable

So verŝajne is stressed on ŝaj:

• ver-ŜAJ-ne

It can move. Esperanto adverbs are not locked into one position.

For example:

• Ŝi verŝajne adoptos la hundidon.
• Verŝajne ŝi adoptos la hundidon.
• Ŝi adoptos la hundidon verŝajne.

All are possible, though some placements sound more natural than others depending on emphasis. Usually, verŝajne is placed near the part of the sentence it modifies.

In a normal sentence, yes, Esperanto usually states the subject explicitly.

So if you want to say she will adopt the puppy, you normally include ŝi.

Esperanto verbs do not change form for different persons the way English does, so adoptos by itself only means will adopt. It does not tell you who the subject is. That is why ŝi is useful and normally necessary here.