Post tempestatem pueri ligna sicca quaerunt, quia ignis sine lignis diu manere non potest.

Because post is a preposition that takes the accusative when it means after.

So:

• post = after
• tempestas = storm
• tempestatem = storm in the accusative singular

Together, post tempestatem means after the storm.

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

Because pueri is nominative plural, the form that marks it as the subject of quaerunt.

• puer = boy
• pueri can mean either
  
  • of the boy (genitive singular), or
  • boys (nominative plural)

Here it must be nominative plural, because the verb quaerunt is third person plural: they seek / they look for.

So pueri quaerunt = the boys look for.

Latin often uses lignum / ligna in the plural to mean pieces of wood, logs, or firewood.

• lignum = a piece of wood, wood, timber
• ligna = pieces of wood / firewood

So although English may use the mass noun wood, Latin is imagining actual separate bits of wood to collect for a fire.

That is why ligna is very natural here.

Because it agrees with ligna.

In Latin, adjectives must agree with the nouns they describe in:

• gender
• number
• case

Here:

• ligna is neuter plural accusative
• so the adjective must also be neuter plural accusative
• that gives sicca = dry

So ligna sicca means dry wood or more literally dry pieces of wood.

Because Latin word order is much freer than English word order. An adjective can come before or after its noun.

So both of these are possible in principle:

• sicca ligna
• ligna sicca

Both mean dry wood.

Sometimes word order gives a slight emphasis or stylistic effect, but in a basic sentence like this, the main point is that the endings show the relationship clearly. English depends much more on position; Latin depends much more on endings.

Quaerunt is present tense, third person plural.

It comes from quaerere, meaning to seek, to look for, or to search for.

So:

• quaero = I look for
• quaerit = he/she looks for
• quaerunt = they look for

In this sentence, pueri ligna sicca quaerunt means the boys are looking for dry wood or the boys look for dry wood, depending on context.

Quia means because and introduces a subordinate clause giving the reason.

Main clause:

• Post tempestatem pueri ligna sicca quaerunt
  = After the storm, the boys look for dry wood.

Reason clause:

• quia ignis sine lignis diu manere non potest
  = because a fire cannot last long without wood.

So quia works much like English because.

Because ignis is the subject of potest.

In the clause:

• ignis ... manere non potest

the thing that cannot remain is the fire, so ignis is nominative singular.

• ignis = fire
• potest = is able / can

So literally: fire cannot remain.

Because sine is a preposition that regularly takes the ablative and means without.

So:

• sine = without
• lignis = with wood in the ablative plural

Together:

• sine lignis = without wood

This is another preposition you normally memorize with its case:
sine + ablative.

For the same reason that ligna was plural earlier: Latin often thinks of firewood as multiple pieces of wood.

So:

• sine lignis = without wood / without pieces of wood / without firewood

English often uses wood as a mass noun, but Latin commonly uses the plural when talking about fuel or gathered wood.

Diu is an adverb meaning for a long time or long.

So:

• diu manere = to remain for a long time, to last long

Its position is flexible. Latin adverbs can often move around more freely than English adverbs. Here it comes before manere, but Latin could arrange the words differently without changing the basic meaning.

Because it depends on potest.

In Latin, possum (I am able, I can) is commonly followed by an infinitive:

• potest manere = can remain
• literally, is able to remain

So in the sentence:

• ignis sine lignis diu manere non potest

the core structure is:

• ignis = the fire
• non potest = cannot
• manere = remain / stay / last

Because its subject is ignis, which is singular.

• ignis = fire
• potest = can, is able (third person singular)

If the subject were plural, you would expect a plural verb, such as possunt.

So Latin is matching the verb to the subject just as English does:

• the fire cannot
• not the fire cannots or the fires cannot unless the subject were plural

Latin can rearrange it much more freely than English, because the endings show what each word is doing.

For example, the sentence could be varied in several ways without changing the basic meaning very much. The exact style or emphasis might shift, but the grammar would still be clear because of the cases and verb endings.

For instance:

• Pueri post tempestatem ligna sicca quaerunt
• Quia ignis diu sine lignis manere non potest, pueri ligna sicca quaerunt

English usually needs stricter order to show meaning. Latin relies more on:

• case endings
• verb endings
• agreement

So the order here is natural, but not the only possible one.

Latin does not have articles like English a and the.

So ignis by itself can mean:

• fire
• a fire
• the fire

Which English version is best depends on context. In this sentence, English might naturally say:

• because a fire cannot last long without wood
• or because fire cannot last long without wood

Latin leaves that unstated, and the reader understands it from the situation.