Babcia prosi, żebyśmy przyjechali ósmego października na rodzinny obiad.

Żebyśmy means so that we / that we / for us to in this kind of sentence.

It is made of:

• żeby = a conjunction often used after verbs of wanting, asking, requesting, etc.
• -śmy = the ending meaning we

So Babcia prosi, żebyśmy... literally means something like Grandma asks that we... or Grandma is asking us to...

After verbs like prosić (to ask/request), Polish normally uses żeby when introducing the action someone wants another person to do.

So:

• Babcia prosi, żebyśmy przyjechali... = Grandma asks us to come...

Using że here would sound wrong, because że usually introduces a statement or fact, not a requested action.

That is a very common learner question.

After żeby, Polish uses a form built from the past-tense stem plus personal endings:

• żebym przyjechał = so that I come
• żebyś przyjechał = so that you come
• żebyśmy przyjechali = so that we come

So even though przyjechali looks like they came, here it does not mean past time. In this structure, it expresses a desired, expected, or requested action.

Because przyjedziemy means we will come/arrive as a normal future statement.

But here the sentence is not simply saying what will happen. It is reporting Grandma’s request. After prosi, żebyśmy..., Polish uses the żeby + past-like form construction:

• przyjedziemy = we will come
• żebyśmy przyjechali = that we come / for us to come

The form agrees with the hidden subject we in żebyśmy.

In Polish plural past-type forms distinguish gender:

• przyjechali = masculine personal plural
• przyjechały = non-masculine-personal plural

If the group includes at least one male, or if the group is mixed/unspecified, Polish normally uses przyjechali.

If the speaker means a group of only women, it would be:

• Babcia prosi, żebyśmy przyjechały...

Przyjechać means to come/arrive by vehicle or more generally to come from somewhere farther away, often by car, train, bus, etc.

So this suggests the people are expected to travel in for the meal.

Compare:

• przyjść = come, arrive on foot / in a general sense
• przyjechać = come by transport

So Grandma is asking them to come over / come in on that date, probably from another place.

Because the sentence refers to one specific completed arrival on one specific date.

• przyjechać is perfective: a single arrival, seen as a completed event
• przyjeżdżać is imperfective: repeated arrivals, habitual action, or focus on the process

Here Grandma is asking for one particular visit:

• żebyśmy przyjechali ósmego października = that we come on the eighth of October

In Polish, when giving a specific calendar date, the day and the month are usually in the genitive.

So:

• ósmy = eighth
• ósmego = of the eighth / on the eighth
• październik = October
• października = of October

That is why Polish says:

• ósmego października = on the eighth of October

This is the normal way to express dates.

Because Polish usually does not need a preposition there.

English says:

• on the eighth of October

Polish normally just says:

• ósmego października

So the idea of on is built into the date expression itself. This often feels strange to English speakers at first, but it is completely normal Polish.

Because na is commonly used with meals and events to mean to/for that occasion.

For example:

• na obiad = for dinner / to dinner / for lunch, depending on context
• na spotkanie = to a meeting
• na wesele = to a wedding

So:

• na rodzinny obiad = for a family dinner / to a family dinner

Here rodzinny obiad is in the accusative after na, but because obiad is masculine inanimate, the accusative looks the same as the nominative:

• nominative: rodzinny obiad
• accusative: rodzinny obiad

It can mean either, depending on context.

Polish present tense often covers both:

• a simple present meaning
• a present continuous meaning

So Babcia prosi could be understood as:

• Grandma asks
• Grandma is asking
• in natural English here, often Grandma is asking us to come...

Yes, Polish often leaves out things that are clear from context.

For example, the sentence does not explicitly say:

• us after prosi
• to her place after przyjechali

But both may be understood from context:

• Babcia prosi already strongly suggests she is asking us
• na rodzinny obiad may imply coming to Grandma’s place or to the family gathering place

A more explicit version could be:

• Babcia prosi nas, żebyśmy przyjechali do niej ósmego października na rodzinny obiad.

Yes. Polish word order is fairly flexible, and changing it can shift emphasis.

For example:

• Babcia prosi, żebyśmy przyjechali ósmego października na rodzinny obiad.
• Babcia prosi, żebyśmy ósmego października przyjechali na rodzinny obiad.
• Babcia prosi, żebyśmy przyjechali na rodzinny obiad ósmego października.

All are understandable, but the original version is a very natural neutral order.

Yes, structurally that is quite close.

A more literal breakdown is:

• Babcia = Grandma
• prosi = asks / requests
• żebyśmy przyjechali = that we come / for us to arrive
• ósmego października = on the eighth of October
• na rodzinny obiad = for a family dinner

But in natural English, you would usually say something like:

• Grandma is asking us to come on October 8 for a family dinner.

So the Polish structure is a little more literal than the most natural English version.