Aan die een kant: standvoëls; maar aan die ander kant: trekvoëls.

It is a fixed contrast pattern, like English On the one hand ... on the other hand ....

• aan die een kant = on the one side / on the one hand
• aan die ander kant = on the other side / on the other hand

Afrikaans uses this to set up a comparison or contrast between two categories.

Aan is a preposition here. In this expression, aan die ... kant literally means something like at/on the ... side.

So:

• aan = at / on / to
• die kant = the side

You should learn aan die een kant and aan die ander kant as whole expressions, rather than translating each word too mechanically every time.

Maar means but.

It marks the contrast between the two sides:

• Aan die een kant: standvoëls
• maar aan die ander kant: trekvoëls

In English, on the one hand ... on the other hand ... sometimes works without but, but in Afrikaans maar often makes the contrast more explicit and natural.

This is an elliptical structure, meaning some words are left out because they are understood from context.

A fuller version might be something like:

• Aan die een kant is daar standvoëls, maar aan die ander kant is daar trekvoëls.

That would mean On the one hand there are resident birds, but on the other hand there are migratory birds.

The shorter version is more compact and works well in headings, summaries, lists, or explanatory text.

Standvoëls is a compound noun:

• stand = staying / remaining in place
• voëls = birds

So standvoëls are birds that stay in one area rather than migrating. In English these are usually called resident birds.

Afrikaans uses compounds very freely, so long words like this are very common.

Trekvoëls is also a compound noun:

• trek = move / migrate / travel
• voëls = birds

So trekvoëls are migratory birds.

The contrast is neat and typical of Afrikaans word-building:

• standvoëls = birds that stay
• trekvoëls = birds that migrate

The ë is a diaeresis. It shows that the vowels are pronounced separately rather than blended into one sound.

• singular: voël
• plural: voëls

The spelling helps you see that the word has separate vowel sounds. This kind of marking is fairly common in Afrikaans.

You also keep that spelling inside compounds:

• standvoëls
• trekvoëls

The most important thing is that the vowels are kept separate because of ë.

Roughly:

• voël sounds somewhat like foo-ul or voo-ul
• voëls adds an s at the end: foo-uls

The exact sound will vary by accent, but the key point is: do not squash it into a single vowel sound.

This is the normal order for a prepositional phrase in Afrikaans:

• aan = preposition
• die een kant = noun phrase

So the structure is:

preposition + article + adjective/number + noun

That is why you get:

• aan die een kant
• aan die ander kant

This order is fixed and natural. A learner should treat both phrases as set expressions.

In this sentence:

• een = one
• ander = other

So:

• die een kant = the one side
• die ander kant = the other side

This pairing is very common in Afrikaans when contrasting two things.

The punctuation gives the sentence a slightly listed, balanced, or formal feel.

• the colon introduces each category
• the semicolon separates the two contrasting parts

So the punctuation visually mirrors the structure:

• one side: standvoëls
• other side: trekvoëls

You could also find similar ideas written with commas in less formal contexts, but this punctuation makes the contrast especially clear.

Yes. A full version would be perfectly good Afrikaans:

• Aan die een kant is daar standvoëls, maar aan die ander kant is daar trekvoëls.

That is more complete and sentence-like.
The original version is shorter and more stylistic, almost like a labeled comparison.

Both are understandable; the shorter one is just more condensed.