Vores nye nabo er meget flink og hjælper gerne med tunge kasser.

Because vores already works as the determiner. In Danish, a possessive like min, din, vores, deres usually replaces an article.

So:

• vores nye nabo = our new neighbor
• not vores en nabo

This is the same basic idea as in English: you say our neighbor, not our a neighbor.

Because adjectives usually take -e when they come after a possessive word like vores.

So:

• en ny nabo = a new neighbor
• vores nye nabo = our new neighbor

This -e form is very common after:

• min
• din
• sin
• vores
• jeres
• deres

You will also often see -e with plural nouns and with definite forms.

Here, flink means something like kind, nice, or helpful.

In Danish, flink is a very common word, but its exact meaning depends on context. It can describe someone who is:

• kind
• pleasant
• helpful
• good or capable in a practical sense

In this sentence, because the neighbor hjælper gerne, flink clearly has the sense of kind/helpful.

Gerne often means gladly, willingly, or likes to.

So hjælper gerne means:

• is happy to help
• helps willingly
• likes to help

It adds the idea that the person is not just helping, but doing so willingly.

This word is extremely common in Danish. For example:

• Jeg kommer gerne. = I’d be happy to come.
• Han læser gerne. = He likes reading / He likes to read.

The preposition med is used with hjælpe when you mean help with something.

So:

• hjælpe med noget = help with something

Examples:

• Hun hjælper med lektierne. = She helps with the homework.
• De hjælper med flytningen. = They help with the move.

In your sentence, med tunge kasser means with heavy boxes.

Because kasser is plural, and adjectives before plural nouns normally take -e.

So:

• en tung kasse = a heavy box
• tunge kasser = heavy boxes

This is one of the most useful adjective patterns in Danish:

• singular, common gender indefinite: often no extra ending
• plural: usually -e

Because this is an indefinite plural expression: heavy boxes in general, not the heavy boxes.

Compare:

• med tunge kasser = with heavy boxes
• med de tunge kasser = with the heavy boxes

So the sentence is talking generally about the kind of thing the neighbor helps with, not a specific set of boxes already known to the listener.

They are both in the present tense.

• er = is
• hjælper = helps

In this sentence, the present tense describes a general characteristic:

• the neighbor is very kind
• and helps willingly

This is similar to English present simple when you describe a person’s usual behavior.

Because the same subject, vores nye nabo, applies to both parts.

The sentence is basically saying:

• Vores nye nabo er meget flink
• and (vores nye nabo) hjælper gerne med tunge kasser

Danish often leaves out the repeated subject when it is obvious and the two parts are closely connected by og.

So the structure is very natural:

• Our new neighbor is very kind and gladly helps with heavy boxes.

Meget means very here. It strengthens the adjective flink.

So:

• flink = kind / nice / helpful
• meget flink = very kind / very nice / very helpful

This is a common pattern in Danish:

• meget god = very good
• meget sød = very sweet / very nice
• meget stor = very big

A simple learner-friendly approximation is YEL-per, but with a Danish æ sound.

A few helpful points:

• hj at the beginning is usually not pronounced like full English h + y; it often sounds closer to just a y sound
• æ is a front vowel somewhat like the vowel in cat, though not exactly the same
• the r in Danish is not like a strong English r

You do not need perfect pronunciation right away. The important thing is to recognize that hjælper is the present tense of hjælpe, meaning to help.

Here are the most useful base forms:

• vores = our
• ny = new
• nabo = neighbor
• være = to be
• meget = very
• flink = kind / nice / helpful
• hjælpe = to help
• gerne = gladly / willingly
• med = with
• tung = heavy
• kasse = box

This can help you see how the sentence is built from dictionary forms plus normal endings such as:

• ny → nye
• tung → tunge
• hjælpe → hjælper
• kasse → kasser