In media via currere non licet, quia plaustra ibi transeunt.

Because in uses different cases depending on the meaning:

• in + ablative = location, meaning in / on / at
• in + accusative = motion toward, meaning into

Here the sentence is talking about where running happens: in the middle of the road. That is location, so Latin uses the ablative:

• media via = ablative singular

If the idea were movement into the road, then you would expect the accusative instead.

Media is an adjective meaning middle or in the middle of. It modifies via, which is a feminine noun meaning road, street, or way.

So the forms match:

• media = feminine singular ablative
• via = feminine singular ablative

Latin adjectives have to agree with the nouns they describe in gender, number, and case.

More or less literally, yes, but in natural English we would say in the middle of the road.

Latin often expresses this idea with an adjective directly modifying the noun:

• media via = the middle of the road / mid-road

So it is normal Latin, even though the most word-for-word English version sounds slightly odd.

Because licet commonly takes an infinitive to say that an action is permitted.

So:

• currere licet = it is permitted to run
• currere non licet = it is not permitted to run

The infinitive currere names the action itself. In English we often do something similar with to run.

Licet means it is allowed, it is permitted, or one may.

It is usually used impersonally, which means there is no personal subject like he or they. The sense is simply:

• licet = it is allowed
• non licet = it is not allowed

So currere non licet is literally something like to run is not permitted.

Non is the normal word for negating a verb in Latin.

So:

• licet = it is allowed
• non licet = it is not allowed

Latin word order is flexible, but placing non before the verb is very common and straightforward.

Plaustra is the nominative plural of plaustrum, a neuter noun meaning cart or wagon.

In this sentence it is the subject of transeunt:

• plaustra = carts
• transeunt = pass / go across

So plaustra ibi transeunt means carts pass there.

Because its subject, plaustra, is plural.

• plaustra = plural
• transeunt = third person plural present active indicative

Even though plaustra is neuter plural, it still normally takes a plural verb in standard prose Latin.

It comes from transeo, transire, transii, transitum, meaning to go across, pass across, or simply pass.

So:

• transeunt = they pass
• here, they refers to plaustra

It is a compound of eo meaning to go plus trans meaning across.

Ibi means there.

It points back to the place already mentioned, namely in media via. So the sentence is saying:

• you must not run there
• because carts pass there

Latin often uses an adverb like ibi to make the connection explicit, even when the place is already understood.

Quia means because. It introduces the reason:

• currere non licet = it is not permitted to run
• quia plaustra ibi transeunt = because carts pass there

This is a very common way to introduce a causal clause in Latin.

Latin word order is much more flexible than English because the endings show the grammatical relationships.

This sentence begins with the place expression:

• In media via = in the middle of the road

That puts the location first for emphasis. Then comes the action that is forbidden:

• currere non licet = it is not allowed to run

Then the reason:

• quia plaustra ibi transeunt = because carts pass there

English relies more heavily on fixed word order, but Latin can move parts around without causing confusion.

Yes. Via can mean road, street, way, or route depending on context.

In this sentence, because carts are passing, road or street is the most natural choice. The exact English word depends on the setting.

Yes. Another common pattern is:

• in medio viae

That literally means in the middle of the road, using medium almost like a noun: the middle.

The sentence you have uses:

• in media via

Both are understandable and natural, but they are built slightly differently. In your sentence, media is simply an adjective agreeing with via.

Literally, it is closer to it is not permitted to run or one may not run.

Depending on context, natural English might turn that into:

• You must not run
• Running is not allowed
• Do not run

So the Latin is impersonal and general, not aimed at a specific you in its grammar, even though English may translate it that way.