Post osculum avia infantem in cubili ponit et culcitam molliorem parat.

Because post as a preposition meaning after takes the accusative case.

So:

• osculum = accusative singular of osculum
• post osculum = after a kiss

This is something you learn with the preposition itself: post + accusative.

The case endings tell you.

• avia is nominative singular, so it is the subject: the person doing the action.
• infantem is accusative singular, so it is the direct object: the one being placed.

So Latin shows who does what to whom mainly by case endings, not just by word order.

Latin normally has no articles.

So:

• avia can mean the grandmother or a grandmother
• infantem can mean the baby/infant or an infant
• culcitam can mean the mattress/cushion or a mattress/cushion

The context decides which English article is best.

In + ablative usually expresses location: in/on the bed, in the crib.

Here:

• cubili is ablative singular
• so in cubili means in the bed/crib or on the bed

A learner might expect in + accusative with a verb like ponit, because that often marks motion into something. That is a reasonable expectation. But in cubili presents the bed more as the place where the baby is laid and ends up.

It comes from cubile, a neuter third-declension noun meaning bed, couch, crib, resting-place.

In this sentence:

• cubili = ablative singular
• it is used after in

So in cubili is a prepositional phrase meaning in/on the bed or crib.

Because molliorem is describing culcitam, so the adjective has to agree with the noun.

• culcitam = accusative singular feminine
• molliorem = accusative singular feminine too

That agreement shows that molliorem belongs with culcitam: a softer mattress/cushion.

Also, culcitam is accusative because it is the direct object of parat.

Molliorem is the comparative form of mollis, meaning soft.

So molliorem means:

• softer

Because it agrees with culcitam, the phrase means a softer mattress/cushion.

Latin often uses a comparative without stating the second thing explicitly, if the context already makes the comparison clear.

So molliorem can simply mean:

• softer
• a softer one
• somewhat soft / rather soft, depending on context

If Latin wanted to state the comparison clearly, it could add:

• quam
  • the thing compared or
• an ablative of comparison

But it does not have to.

The subject carries over from the first verb.

So the sentence means that the grandmother:

• places the baby in/on the bed
• and prepares a softer mattress/cushion

Latin often leaves out a repeated subject when it is already obvious.

Latin word order is fairly flexible, because the case endings already show the grammatical roles.

This sentence puts the words in a natural order:

• Post osculum = time expression first
• avia = subject
• infantem = object
• in cubili = place
• ponit = verb
• then the second action: et culcitam molliorem parat

A different order could still be grammatical, but the emphasis might change. Latin word order often helps show what the writer wants to highlight.