Post tempestatem multum lutum in vico manet, et pueri extra domum ludere non possunt.

Because post takes the accusative case when it means after. So:

• tempestas = storm
• tempestatem = after the storm when used with post

This is a very common pattern: post + accusative.

Because lutum is a neuter singular noun meaning mud, and it is treated as a mass noun, like mud in English. You usually do not count mud as separate items, so Latin uses the singular.

So:

• lutum = mud
• multum lutum = much mud or a lot of mud

multum agrees with lutum, so it is also neuter singular.

Because the subject is lutum, which is singular.

Even though multum lutum means a lot of mud, the head noun is still lutum, and that noun is singular. So the verb must also be singular:

• lutum ... manet = the mud remains

This is similar to English a lot of mud remains, not remain.

It is nominative singular neuter, because it is the subject of manet.

You can see the basic structure like this:

• Post tempestatem = after the storm
• multum lutum = much mud / a lot of mud
• in vico = in the street
• manet = remains

So multum lutum is the thing doing the remaining.

Because in takes:

• the ablative for location: in/at/on
• the accusative for motion into

Here the sentence means the mud is in the street, not moving into it. So Latin uses the ablative:

• in vico = in the street

If it meant something like into the street, you would expect in vicum.

It can, depending on context. Vicus can mean things like:

• village
• street
• neighborhood
• lane

In this sentence, the meaning is something like street or lane, because that fits the idea of mud remaining there after a storm.

Because pueri is the subject of possunt.

• puer = boy
• pueri = boys or the boys

Since the boys are the ones who cannot play, the noun must be in the nominative plural.

Because extra takes the accusative case. So:

• extra domum = outside the house

This is worth noticing because an English speaker might expect a location word to use the ablative, but extra is simply a preposition that governs the accusative.

Also, domus is an irregular noun, and domum is its accusative singular form here.

Because after possum (I am able, I can), Latin normally uses an infinitive.

So:

• possunt ludere = they can play
• non possunt ludere = they cannot play

This is very similar to English can play, except Latin uses the infinitive form ludere.

Because non negates the finite verb phrase.

• possunt = they can
• non possunt = they cannot

Latin often places non directly before the verb it is negating. So pueri extra domum ludere non possunt means the boys cannot play outside the house.

Because Latin has no articles. There is no direct equivalent of English the or a/an.

So a Latin noun like pueri can mean:

• boys
• the boys

And domum can mean:

• a house
• the house

The context tells you which is most natural in English.

Because Latin word order is much more flexible than English word order. Latin relies heavily on case endings, so the job of each word is often clear even when the order changes.

This sentence is arranged in a very natural Latin way:

• Post tempestatem puts the time idea first
• multum lutum introduces the subject
• in vico adds the place
• manet comes at the end of the clause

Then in the second clause:

• pueri = subject
• extra domum = where
• ludere = what they want to do
• non possunt = cannot

So the order may feel unusual in English, but it is normal Latin style.