Дети машут бабушке у входа.

Because махать usually takes the dative case for the person you wave to.

• бабушка = nominative, the basic dictionary form
• бабушке = dative singular, meaning to grandma / to the grandmother

So:

• Дети машут бабушке = The children are waving to grandma

A useful pattern is:

• махать кому? = to wave to whom?

So you get:

• машу маме = I wave to Mom
• машем друзьям = we wave to friends

Because дети is in the nominative case, which is the usual case for the subject of the sentence.

In this sentence:

• Дети = the children, the subject
• машут = wave / are waving
• бабушке = to grandma

So the structure is basically:

• [subject] [verb] [indirect object] [place]

Even though Russian word order is flexible, the case endings help show who is doing what.

Yes, дети means children, and it is the irregular plural connected with ребёнок = child.

This is one of the very common irregular noun patterns in Russian:

• ребёнок = child
• дети = children

So you should just learn дети as a basic high-frequency plural form.

Машут is present tense.

In Russian, the present tense of an imperfective verb can often translate as either:

• wave
• are waving

depending on context.

So Дети машут бабушке could mean:

• The children wave to grandma if you mean something habitual
• The children are waving to grandma if you mean it is happening right now

English forces you to choose more clearly than Russian often does.

The infinitive is махать = to wave.

Its present-tense forms are built with a changed stem:

• я машу
• ты машешь
• он/она машет
• мы машем
• вы машете
• они машут

So машут is simply the correct 3rd person plural form, agreeing with дети.

This kind of stem change is something you just have to learn with the verb. Russian verbs do this fairly often.

Because у with the genitive case often means by, near, or at something.

Here:

• вход = entrance
• входа = genitive singular
• у входа = by the entrance / at the entrance

So Дети машут бабушке у входа means the waving happens near the entrance.

This is different from в + prepositional, which usually means in or inside:

• в доме = in the house
• у дома = by the house

Because the preposition у requires the genitive case.

So:

• вход = nominative
• у входа = genitive after у

This is a very important Russian pattern. Some common examples:

• у окна = by the window
• у двери = by the door
• у школы = by the school

So in this sentence, у входа is exactly what you should expect grammatically.

Yes. Russian word order is much more flexible than English word order because the case endings carry a lot of the grammatical information.

The neutral order here is:

• Дети машут бабушке у входа.

But you could also hear:

• У входа дети машут бабушке.
• Бабушке дети машут у входа.

These versions shift the emphasis a bit:

• starting with у входа emphasizes the location
• starting with бабушке emphasizes who they are waving to

Even when the word order changes, the cases still tell you the roles:

• дети = subject
• бабушке = indirect object

Not always, but very often.

Бабушка literally means grandmother or grandma, but in some contexts it can also refer to an elderly woman.

So depending on context, бабушке could mean:

• to grandma
• to the grandmother
• sometimes to an elderly lady

If the meaning has already been given to you, then you should follow that meaning. But it is good to know the word can be a little broader in real usage.

Most naturally, it describes the location of the whole scene: the waving is happening at/by the entrance.

In many real contexts, that could imply:

• the children are at the entrance
• the grandmother is at the entrance
• both are near the entrance

Russian often leaves this kind of detail to context unless the speaker wants to be more specific.

If you wanted to be clearer, you could say something more explicit, for example:

• Дети у входа машут бабушке. = The children at the entrance are waving to grandma.
• Дети машут бабушке, стоящей у входа. = The children are waving to grandma, who is standing at the entrance.

Because Russian has no articles like a, an, or the.

So:

• дети can mean children or the children
• бабушке can mean to grandma, to a grandmother, or to the grandmother
• у входа can mean at an entrance or at the entrance

Context tells you which English article is most natural.

That is why a short Russian sentence can sometimes have several possible English translations, even though the Russian itself is perfectly normal and clear in context.