Afløbet i køkkenet er stoppet, så jeg må ringe til en håndværker.

Because Danish usually puts the definite article at the end of the noun. Both afløb and køkken are neuter nouns:

et afløb = a drain
et køkken = a kitchen

So the definite forms are:

afløbet = the drain
køkkenet = the kitchen

i usually means in, and with rooms Danish often uses i just like English does: in the kitchen, in the living room, and so on.

So afløbet i køkkenet means the drain in the kitchen.

Here er stoppet means is blocked or is clogged.

Even though stoppet comes from the verb at stoppe, in this sentence it works like a descriptive word after er. So it is not really has stopped in the English sense. For a drain, the natural English meaning is is clogged/blocked.

Yes. Stoppet is very common in everyday Danish for a blocked drain. You may also hear tilstoppet, which is even more specific and literally means blocked up.

So both of these are natural:

Afløbet er stoppet.
Afløbet er tilstoppet.

Here så means so, therefore, or as a result.

It links the two ideas:

The drain is clogged, so I have to call a tradesperson.

Because the sentence contains two full clauses:

Afløbet i køkkenet er stoppet
jeg må ringe til en håndværker

In standard Danish writing, a comma is normally used before a conjunction like så when it joins two independent clauses.

Because this så is a coordinating conjunction meaning so. After a coordinating conjunction, Danish keeps normal main-clause word order:

jeg må ringe

So the subject jeg comes before the verb må.

In this sentence, må means must or have to.

This often confuses English speakers because it looks similar to may, but Danish må very often expresses necessity:

Jeg må ringe = I have to call

Because at ringe normally takes the preposition til when it means to call someone on the phone.

So:

ringe til nogen = call someone

That is why Danish says ringe til en håndværker, not normally ringe en håndværker.

Håndværker is a general word for a tradesperson or craftsperson: someone like a plumber, electrician, carpenter, and so on.

In this sentence, it is left general. Since the problem is a drain, a plumber is probably what is meant in real life, but the Danish sentence does not specify that.

Because the speaker means a tradesperson, not the tradesperson.

en håndværker = any suitable tradesperson
håndværkeren = a specific, already known tradesperson

So the indefinite form is used because no particular person has been identified.

They do not match English sounds exactly, but roughly:

ø in afløbet and køkkenet is similar to a vowel found in French or German, not in normal English.
å in må and håndværker is roughly like the vowel in law for many English speakers.

The best way to learn them is by listening to native pronunciation, because English does not have exact equivalents.