Namate die verkeer besiger raak, ry sy stadiger.

Namate means as, in proportion as, or the more... the more... when something changes gradually.

So Namate die verkeer besiger raak... means something like:

• As the traffic gets busier...
• The busier the traffic gets...

It is not just a simple time word like when. It specifically links two changes that happen together.

Because namate introduces a subordinate clause.

In Afrikaans, the finite verb usually goes to the end of a subordinate clause:

• Namate die verkeer besiger raak

Compare that with a main clause:

• Die verkeer raak besiger

So the word order changes because of namate.

Literally, besiger raak means become busier or get busier.

• besig = busy
• besiger = busier
• raak = become / get

So:

• die verkeer besiger raak = the traffic gets busier

Afrikaans often forms the comparative with -er, especially with common shorter words.

So:

• besig → besiger
• stadig → stadiger

That is the normal, natural pattern here.
Using meer besig would sound much less idiomatic in this sentence.

Because the sentence begins with another clause: Namate die verkeer besiger raak.

When something other than the subject comes first in a main clause, Afrikaans usually puts the verb before the subject. This is often called inversion.

So:

• normal order: Sy ry stadiger
• after the first clause: ..., ry sy stadiger

That is why ry comes before sy.

Here it means she.

Even though sy can sometimes correspond to her in English, in this sentence it is the subject of the verb ry:

• ry sy stadiger = she drives more slowly

Because of the word order after the first clause, the subject comes after the verb, but it is still the subject.

Ry can mean drive, ride, or sometimes travel by vehicle, depending on context.

In this sentence, because it talks about traffic, the most natural meaning is drive:

• ry sy stadiger = she drives more slowly

For the same reason as besiger: Afrikaans commonly uses -er to make the comparative.

• stadig = slow / slowly
• stadiger = slower / more slowly

So ry sy stadiger means she drives more slowly or she drives slower.

Here it functions as an adverb, because it describes how she drives.

• stadiger modifies ry

A useful thing to remember is that Afrikaans does not have a separate -ly ending like English. So the same form often covers both adjective and adverb uses.

That is why stadiger can mean either:

• slower
  or
• more slowly

depending on context.

Because verkeer is a mass/collective noun, like English traffic.

So even though traffic involves many cars, the word itself is grammatically treated as one whole idea:

• die verkeer = the traffic

That is why the sentence does not treat it like a plural noun such as cars.

Yes. A very similar idea can be expressed as:

• Hoe besiger die verkeer raak, hoe stadiger ry sy.

This means:

• The busier the traffic gets, the more slowly she drives.

So namate and hoe ... hoe ... can both express linked gradual change, though the structure is a little different.