Mense Ianuario nix saepe cadit, et mater pueris dicit Ianuarium frigidissimum mensem anni esse.

Because mense Ianuario is in the ablative case, which Latin often uses for time when.

So:

• mense = in the month
• Ianuario = January, agreeing with mense

Together they mean in January, literally in the month of January.

If you changed them to mensis Ianuarius, that would be nominative, which would make them the subject, not a time expression.

Yes. Latin often does that.

• Ianuario can mean in January
• mense Ianuario is a fuller, more explicit way to say it

So both are possible, but mense Ianuario makes the time expression especially clear.

Latin has no articles like English the, a, or an.

So when you translate, English adds them where needed:

• mater = mother or the mother
• pueris = to the children/boys
• frigidissimum mensem anni = the coldest month of the year

The Latin sentence does not contain separate words for those articles.

Latin often expresses this idea more concretely:

• nix = snow
• cadit = falls

So nix cadit means snow falls.

English often uses an impersonal expression, it snows, but Latin can simply use snow as the subject. That is very natural Latin.

Cadit is:

• 3rd person singular
• present tense
• from cadere = to fall

It matches nix, which is singular.

So:

• nix cadit = snow falls
• saepe cadit = often falls

Saepe is an adverb, meaning often.

It modifies cadit:

• nix saepe cadit = snow often falls

Latin word order is more flexible than English word order, so saepe could appear in different places without changing the basic meaning. Its position here is normal and natural.

Because pueris is the indirect object of dicit.

• mater = the subject
• dicit = says
• pueris = to the children/boys

So mater pueris dicit means the mother says to the children.

This is a very common Latin pattern:

• dicere alicui = to say to someone

Because after verbs of saying, thinking, knowing, and similar verbs, Latin very often uses an indirect statement.

Instead of English she says that January is..., Latin uses:

• accusative + infinitive

Here that means:

• Ianuarium = subject of the indirect statement, in the accusative
• esse = infinitive, to be
• frigidissimum mensem anni = predicate, the coldest month of the year

So literally Latin says something like:

• the mother says January to be the coldest month of the year

That is the normal Latin way to express she says that January is the coldest month of the year.

Because this is an indirect statement.

In Latin indirect statement:

• the subject goes into the accusative
• the verb becomes an infinitive

So:

• Ianuarium = accusative subject of esse
• mensem = predicate accusative, because it refers to the same thing as Ianuarium

English says:

• January is the coldest month

Latin turns that into:

• Ianuarium ... mensem ... esse

Both nouns are accusative because both belong to the indirect statement.

Anni is the genitive singular of annus, meaning year.

It depends on mensem:

• mensem anni = month of the year

So:

• frigidissimum mensem anni = the coldest month of the year

This genitive is very common in Latin for showing a relationship like of the year.

Frigidissimum is the superlative of frigidus:

• frigidus = cold
• frigidior = colder
• frigidissimus = coldest

Here it is:

• masculine
• accusative
• singular

It agrees with mensem, so:

• frigidissimum mensem = the coldest month

Since Ianuarium and mensem refer to the same thing, the whole phrase means January to be the coldest month.

Latin often puts the verb, especially an infinitive like esse, near the end of the clause.

So:

• Ianuarium frigidissimum mensem anni esse

is a very typical Latin order.

But Latin word order is flexible, so the exact position can vary. The ending -m on Ianuarium, frigidissimum, and mensem, plus the infinitive esse, tells you how the words fit together even if the order is not the same as English.