Čím déle čekáme na obsluhu, tím větší hlad má moje babička.

It is the Czech way to say the ..., the ... in comparisons.

So:

• Čím déle čekáme ..., tím větší hlad ... = The longer we wait ..., the hungrier / the greater hunger ...

A few more examples:

• Čím víc čtu, tím víc rozumím.
  = The more I read, the more I understand.
• Čím je starší, tím je klidnější.
  = The older he/she is, the calmer he/she is.

This is a very common structure in Czech.

Historically, they are forms of co and to, but in this pattern you should learn them as a fixed pair:

• čím = the more / the longer / the less ...
• tím = the more / the bigger / the less ...

So in this sentence:

• čím déle = the longer
• tím větší = the bigger / the greater

You do not need to translate them word-for-word as by what and by that here. In this construction, they function together as a set expression.

Déle is the comparative form of dlouho (for a long time / long in the adverbial sense).

• dlouho = long
• déle = longer

So:

• čekáme dlouho = we are waiting a long time
• čekáme déle = we are waiting longer

This comparative is a little irregular, so it is worth memorizing:

• dlouho → déle

In informal spoken Czech, you may also hear dýl, but déle is the standard form.

Yes, literally it means bigger/greater hunger, and that is a very natural Czech way to express the idea.

Czech often uses:

• mít hlad = to be hungry
• mít větší hlad = to be hungrier / to have more hunger

So even though English usually says hungrier, Czech commonly says have a bigger hunger.

You can also say:

• Čím déle čekáme, tím je moje babička hladovější.

That means essentially the same thing, but mít hlad / mít větší hlad is very idiomatic and common.

Because Czech normally expresses this idea with the noun hlad (hunger) and the verb mít (to have):

• Mám hlad. = I am hungry.
• Máš hlad? = Are you hungry?
• Moje babička má hlad. = My grandmother is hungry.

So this is one of those places where Czech and English structure the idea differently:

• English: to be hungry
• Czech: to have hunger

This is completely normal and very common.

In standard modern Czech, čekat usually takes na + accusative when it means wait for someone or something.

So:

• čekat na vlak = to wait for the train
• čekat na kamaráda = to wait for a friend
• čekat na obsluhu = to wait for the service / waitstaff

That is why you see na obsluhu here.

Here obsluha means something like:

• the waitstaff
• the service
• the person serving us

In a restaurant context, čekat na obsluhu means waiting for someone to come serve you.

So it does not mean machine operation here, even though obsluha can also mean operation or operator/service personnel in other contexts. The restaurant context makes the meaning clear.

Because na with čekat na requires the accusative case.

The dictionary form is:

• obsluha = nominative singular

But after na in this sentence, it becomes accusative singular:

• na obsluhu

This is a regular feminine noun pattern.

Compare:

• ta obsluha = the service / the waitstaff
• čekáme na obsluhu = we are waiting for the service / waitstaff

It actually is the object of má, so it is functioning as an accusative. But hlad is a masculine inanimate noun, and in the singular its nominative and accusative forms are the same.

So:

• nominative: hlad
• accusative: hlad

That is why you do not see any visible change.

This is similar to many masculine inanimate nouns in Czech.

Yes, you can say moje babička má too.

Czech word order is flexible, and the choice often depends on emphasis, rhythm, or what information feels more important.

The given sentence:

• Čím déle čekáme na obsluhu, tím větší hlad má moje babička.

is very natural.

But these are also possible:

• Čím déle čekáme na obsluhu, tím větší hlad moje babička má.
• Čím déle čekáme na obsluhu, tím moje babička má větší hlad.

They are grammatical, but the original version sounds especially smooth and natural.

A useful rule for learners: in Czech, word order often changes the focus, not the basic meaning.

Yes.

Both mean my grandmother:

• moje babička
• má babička

The short form má is perfectly correct. The longer form moje is also very common, especially in everyday speech.

So in this sentence, both are possible:

• ... má moje babička
• ... má má babička

The version with moje babička may sound a bit more natural to many learners at first, because it is easier to hear clearly and is very common in conversation.

Yes, normally yes.

The two parts of the čím ..., tím ... construction are separated by a comma:

• Čím déle čekáme na obsluhu, tím větší hlad má moje babička.

This matches the fact that the sentence has two linked clauses. In written Czech, the comma is standard here.