Когда я вижу их у входа в театр, я говорю «Здравствуйте».

Here когда literally means when, but in this kind of sentence it often has the sense of whenever.

So:

• Когда я вижу их у входа в театр, я говорю …
  = When / Whenever I see them at the entrance to the theater, I say …

Because both verbs are in the present tense, the sentence describes a repeated or habitual situation, not just one single event.

Russian often uses the present tense to describe a habit or repeated action, just like English can:

• When I see them, I say hello.

So я вижу here means I see in a general, repeated sense.

Also, вижу is from the imperfective verb видеть, which is the normal choice for ongoing, repeated, or habitual seeing.

Because увидеть / увижу is perfective, and perfective verbs usually refer to a single completed event in the future.

Compare:

• Когда я вижу их... = whenever I see them / when I am in the habit of seeing them
• Когда я увижу их... = when I see them one day / when I do see them in the future

In your sentence, the meaning is habitual, so вижу is the natural choice.

Их can mean both:

• them
• their

The meaning depends on grammar and context.

Here it means them because it is the direct object of вижу:

• я вижу их = I see them

If it meant their, it would usually be followed by a noun:

• их театр = their theater
• их друзья = their friends

So in this sentence, их clearly means them.

Because the preposition у requires the genitive case.

• dictionary form: вход
• genitive singular: входа

So:

• у входа = by / near the entrance

This is a very common pattern:

• у дома = by the house
• у двери = by the door
• у входа = by the entrance

This is a very common learner question.

In у входа в театр, the phrase в театр goes with вход:

• вход в театр = entrance to the theater

So the structure is:

• у входа = by the entrance
• в театр = to the theater

This is not the same as saying in the theater. If you said в театре, that would mean inside the theater.

So:

• у входа в театр = at the entrance to the theater
• в театре = in the theater

Yes, Russian could leave it out:

• Когда я вижу их у входа в театр, говорю Здравствуйте.

That is possible, because the verb ending already shows the subject.

But repeating я is also completely natural. It can make the sentence clearer, more balanced, or a little more emphatic:

• Когда я вижу их..., я говорю...

So the repeated я is not required, but it sounds normal.

Because Когда я вижу их у входа в театр is a subordinate clause, and Russian normally separates subordinate clauses from the main clause with a comma.

Structure:

• subordinate clause: Когда я вижу их у входа в театр
• main clause: я говорю Здравствуйте

So the comma is required.

This is very standard in Russian:

• Когда я прихожу домой, я ужинаю.
• Если он звонит, я отвечаю.

Because говорю fits a repeated, habitual action.

• говорю = I say / I am saying
• скажу = I will say, one completed future utterance

In this sentence, the meaning is:

• Whenever I see them, I say hello.

That is a repeated action, so говорю is the right choice.

If you used скажу, it would sound more like one future occasion:

• Когда я увижу их, я скажу Здравствуйте.
  = When I see them, I will say hello.

Здравствуйте is a polite or formal greeting, roughly hello in a respectful register.

Russian greetings depend a lot on formality:

• Здравствуйте = polite/formal
• Привет = informal, used with friends, family, close peers

So this sentence suggests that the speaker is being polite to these people.

This word is famous for being hard for learners.

A careful pronunciation is roughly:

• ZDRA-stvooy-tye

But in real speech, some sounds are often reduced, and it may sound closer to:

• ZDRAS-tvuy-tye

A few helpful points:

• the в is often not heard very clearly
• the ending -те is the polite/plural ending
• in casual speech, people may shorten it to здрасьте, but that is more colloquial

You do not need to pronounce every letter very heavily to sound natural.

Russian normally uses « » as quotation marks in writing.

So instead of English-style quotation marks, Russian often writes direct speech or quoted words like this:

• Я говорю «Здравствуйте».

These are called guillemets.

Yes. Russian word order is flexible, though not random.

The neutral version here is:

• Когда я вижу их у входа в театр, я говорю Здравствуйте.

But other orders are possible depending on emphasis. For example:

• У входа в театр, когда я вижу их, я говорю Здравствуйте.

That sounds more marked and shifts the focus.

For learners, the original version is a very good standard pattern:

• time clause first
• main clause second

In в театр, театр is in the accusative singular.

That is because в with motion toward a place takes the accusative:

• идти в театр = to go to the theater
• вход в театр = entrance to the theater

Compare that with location:

• в театре = in the theater

So:

• в театр = accusative, direction
• в театре = prepositional, location