Na kružnom toku kraj škole jutros je jedna žena stala da propusti pješaka s djetetom.

Because na is being used here to mean at/on the roundabout as a location, it takes the locative case.

• dictionary form: kružni tok
• locative singular: kružnom toku

So:

• na kružnom toku = at the roundabout

This is a very normal Croatian way to express location with na.

Because kraj as a preposition, meaning near / by / next to, takes the genitive case.

• dictionary form: škola
• genitive singular: škole

So:

• kraj škole = near the school

You can think of kraj here as behaving like a preposition that always wants genitive.

This is because je is a clitic in Croatian. Clitics usually go in the second position of the clause, but that means the second unit, not necessarily the second single word.

In this sentence, the opening adverbial part:

• Na kružnom toku kraj škole jutros

acts as the first chunk of information, and then je comes after it:

• Na kružnom toku kraj škole jutros je jedna žena stala...

Croatian word order is flexible, so you could also see other natural versions, such as:

• Jedna žena je jutros stala...
• Jutros je jedna žena stala...

The version in your sentence sounds natural and slightly narrative.

It can do both jobs.

Croatian has no articles, so jedan / jedna / jedno is often used where English would simply say a / an, especially when introducing something new into the story.

So here jedna žena most naturally means:

• a woman

But it still carries a faint sense of one particular woman or one woman more than English a woman does.

Croatian past tense is usually made with:

• the auxiliary biti in the present tense
• plus the past participle of the main verb

Here:

• je = has / is auxiliary, 3rd person singular
• stala = past participle of stati

Together:

• je stala = stopped

The participle agrees with the subject:

• stao for masculine
• stala for feminine
• stalo for neuter

Because the subject is žena, it must be stala.

This is an aspect question.

• stati is perfective: to stop, as a completed action
• stajati is usually imperfective: to stand, be standing, or sometimes to keep stopping depending on context

In this sentence, the woman performed one completed action:

• she stopped in order to let someone pass

So stala is the correct choice.

If you used stajala, it would suggest more of a state like was standing, which is not the idea here.

Because da + present tense is very commonly used in Croatian to express purpose or an intended action.

So:

• stala da propusti... = she stopped to let ... pass

Here:

• da introduces the purpose
• propusti is the present-tense form used in that da clause

This is extremely natural Croatian. English often uses an infinitive, but Croatian very often uses da + present instead.

Yes, formally it is present tense, 3rd person singular, from the verb propustiti.

After da, Croatian often uses a present-tense form where English would use:

• an infinitive
• or a purpose construction

So da propusti does not mean she is presently letting someone through right now in the same way an independent present tense would. Here it means:

• so that she could let
• to let

Also, propustiti is perfective, which fits a single completed event: allowing the pedestrian to go first.

Because pješak is the direct object of propusti, so it must be in the accusative case.

Since pješak is a masculine animate noun, its accusative singular matches the genitive form:

• nominative: pješak
• accusative: pješaka

This is a very important Croatian pattern:

• masculine animate nouns usually change in the accusative
• masculine inanimate nouns usually do not

Because s meaning with takes the instrumental case.

• dictionary form: dijete
• instrumental singular: djetetom

So:

• s djetetom = with a child

This tells you the pedestrian was accompanied by a child.

In this sentence, it most naturally describes the pedestrian:

• propusti pješaka s djetetom = she let the pedestrian with a child pass

Grammatically, Croatian sometimes allows small attachment ambiguities like English does, but in normal reading this phrase is understood as belonging to pješaka, not žena.

Jutros specifically means this morning.

That is slightly more specific than a general phrase like ujutro, which means in the morning in a broader sense.

So:

• jutros = this morning
• ujutro = in the morning

In your sentence, jutros places the event on the morning of the day being talked about.