Mater vicinas benigne accipit et eis gratulatur, quod filias suas tam diligenter curant.

Because accipit is a transitive verb and takes a direct object in the accusative.
So:

• mater = the subject
• vicinas = the direct object of accipit

In other words, the mother receives/welcomes whom? The neighbors.

Because gratulor takes the person being congratulated in the dative. So eis means to them.

That gives this pattern:

• eis gratulatur = she congratulates them

This is different from English, where we simply say congratulates them without a special case form.

Because gratulor, gratulari is a deponent verb. Deponent verbs use passive-looking forms but have active meanings.

So gratulatur looks like he/she is congratulated, but it actually means he/she congratulates.

That is a very common thing in Latin, and learners often have to get used to it.

Here quod introduces a clause explaining the reason for the congratulation. It can be understood as because or sometimes that in English.

So:

• eis gratulatur, quod... = she congratulates them because...
• or more literally, she congratulates them that...

In smoother English, because is usually best here.

The subject is understood from the verb ending. Curant is third-person plural, so it means they care for / they look after.

From the context, they refers to vicinae—the neighbors. Latin often leaves subject pronouns unstated when the verb ending already makes the person and number clear.

Because curant takes a direct object. In Latin, curare can mean to care for, to look after, or to take care of, and the person cared for is put in the accusative.

So:

• filias suas curant = they take care of their daughters

This can feel a little unusual to English speakers because English uses care for with a preposition, but Latin uses a direct object here.

Suas is the reflexive possessive adjective, meaning their own here. It normally refers back to the subject of its own clause.

In the clause quod filias suas tam diligenter curant, the subject of curant is they = the neighbors. So suas refers to the neighbors’ own daughters.

If Latin wanted to say someone else’s daughters, it would use a different possessive, such as earum or eius, depending on the meaning.

Diligenter means carefully, diligently, or attentively.
Tam means so.

Together:

• tam diligenter = so diligently, so carefully

Tam strengthens the adverb, showing degree.

Latin word order is much more flexible than English word order because the case endings show how the words function.

For example:

• Mater vicinas benigne accipit
• Mater benigne vicinas accipit
• Vicinas mater benigne accipit

All of these could mean roughly the same thing, because mater is nominative and vicinas is accusative.

In this sentence, the order helps the sentence flow naturally, but the endings are what really tell you who is doing what.

Yes:

• accipit = she receives / welcomes
• gratulatur = she congratulates
• curant = they care for / look after

All three are present indicative. Latin often uses the present exactly where English would also use the present.