Prošle zime i mama i ja imale smo gripu, pa smo obje ostale kod kuće.

Because the sentence is spoken by a female speaker about two female people: mama and ja.

In Croatian past tense, the main past participle agrees in gender and number with the subject:

• imale smo = we had, feminine plural
• ostale = stayed, feminine plural

So:

• i mama i ja imale smo gripu = both my mom and I had the flu
• smo obje ostale kod kuće = we both stayed home

If the speaker were male, you would normally say:

• i mama i ja imali smo gripu, pa smo oboje ostali kod kuće

So this sentence strongly suggests that the speaker is female.

Croatian past tense is made with:

• the auxiliary biti in the present tense
• plus the past participle

Here:

• smo = we are/have as an auxiliary
• imale = past participle
• ostale = past participle

In Croatian, short unstressed words like sam, si, je, smo, ste, su are clitics, and they usually go in the second position of the clause.

So instead of:

• i mama i ja smo imale gripu (possible in some contexts, but less neutral here)

you often get:

• i mama i ja imale smo gripu

Likewise:

• pa smo obje ostale kod kuće

The position of smo is mainly about Croatian clitic placement, not emphasis.

i mama i ja means both mom and I or literally both mom and me / mom and I too.

The repeated i ... i ... structure means both ... and ...:

• i mama i ja = both my mom and I
• i Marko i Ana = both Marko and Ana

This is a very common Croatian pattern.

So the sentence could be understood as:

• Both my mom and I had the flu...

prošle zime means last winter.

Here zime is in the genitive singular, used in a common time expression. Croatian often uses the genitive for expressions like:

• prošle zime = last winter
• prošlog ljeta = last summer
• prošle godine = last year

The adjective also agrees with the noun:

• prošle matches zime in gender, number, and case

So this is not a plural meaning like winters here. It is a fixed time expression meaning during last winter.

Because gripa is the direct object of imati (to have), so it goes into the accusative case.

Dictionary form:

• gripa = flu

Accusative singular:

• gripu

So:

• imati gripu = to have the flu

This is very normal Croatian object case usage.

Here pa means something like:

• so
• and so
• therefore
• sometimes simply and then

In this sentence:

• Prošle zime i mama i ja imale smo gripu, pa smo obje ostale kod kuće.

it links the two ideas naturally:

• We had the flu, so we both stayed at home.

It sounds conversational and very natural in Croatian.

obje means both, specifically for two feminine people/things.

Since the two people are female:

• mama
• ja (female speaker)

the correct form is obje.

Compare:

• oba = both, masculine/neuter
• obje = both, feminine
• oboje = often used for mixed gender or as a collective form

So in this sentence:

• smo obje ostale = we both stayed

Everything matches the feminine plural idea of the subject.

kod kuće is the usual idiomatic way to say at home in Croatian.

So:

• ostati kod kuće = stay at home

By contrast:

• u kući means in the house

That refers more literally to being inside the building.

So:

• kod kuće = at home, at one’s home
• u kući = in the house

In your sentence, the idea is the general everyday one: they stayed home because they had the flu. That is why kod kuće is the natural choice.

Yes. You could say:

• Prošle zime mama i ja imale smo gripu...

That simply means Mom and I had the flu last winter.

But:

• i mama i ja

adds the sense of both my mom and I, often with a slight emphasis that the same thing happened to both of us.

So the version in your sentence is a bit more expressive and explicit.

A rough breakdown is:

• Prošle zime = last winter
• i mama i ja = both mom and I
• imale smo = had we
• gripu = the flu
• pa = so
• smo obje ostale = we both stayed
• kod kuće = at home

Very literally:

• Last winter both mom and I had the flu, so we both stayed at home.

The Croatian word order is normal and natural, even if it does not line up perfectly with English word-for-word.

Yes. mama is a feminine noun, and it behaves as feminine in agreement.

That matters because Croatian agreement depends on the actual grammatical gender of the subject. Since the subject here is two female people, the participles are feminine plural:

• imale
• ostale
• obje

So even though English does not show this kind of agreement, Croatian does, and it is an important clue to who is being talked about.