Veterinären behandlar tassen för att slippa operera.

Because veterinären is the definite singular form: the veterinarian / the vet.

• en veterinär = a veterinarian
• veterinären = the veterinarian

The ending -en is the usual definite ending for many common-gender nouns in Swedish.

For the same reason: tassen is the definite singular form, meaning the paw.

• en tass = a paw
• tassen = the paw

In this sentence, a specific paw is meant, not just any paw.

Behandlar is the present tense of behandla, which means to treat.

• behandla = to treat
• behandlar = treats / is treating

So Veterinären behandlar tassen means The vet treats the paw or The vet is treating the paw, depending on context.

För att means to, in order to, or so as to when it introduces a purpose.

So:

• Veterinären behandlar tassen för att slippa operera.

means roughly:

• The vet treats the paw in order to avoid operating.

This is a very common Swedish structure:

• Jag tränar för att må bättre. = I exercise to feel better.
• Hon sparar pengar för att resa. = She saves money to travel.

Slippa means to avoid, to not have to, or to get out of having to do something.

It often has the sense of being spared something unpleasant or unnecessary.

Examples:

• Jag slipper jobba i morgon. = I don’t have to work tomorrow.
• Vi tog bussen för att slippa gå. = We took the bus to avoid walking.

So here, slippa operera means avoid having to operate.

Because after slippa, Swedish normally uses the bare infinitive without att.

So:

• slippa operera = correct
• slippa att operera = usually not the normal choice here

This is similar to other verbs that take an infinitive without att, though it is something you mostly learn verb by verb.

Grammatically, it is the same subject as in the main clause: veterinären.

So the sentence means that the vet treats the paw in order to avoid having to operate.

That is an important point for English speakers: Swedish often leaves the subject of the infinitive clause unspoken when it is the same as the main subject.

So the structure is understood as:

• Veterinären behandlar tassen för att veterinären ska slippa operera.

But Swedish normally does not repeat veterinären there.

Yes, a learner might feel that way, and that is a reasonable reaction.

The grammar is clear enough: the vet wants to avoid operating. But operera has no object stated, so the sentence leaves unsaid exactly what would otherwise be operated on. We understand it from context: probably the paw, the animal, or the injury.

A more explicit version could be:

• Veterinären behandlar tassen för att slippa operera den.
  = The vet treats the paw to avoid operating on it.

Or:

• Veterinären behandlar tassen för att undvika en operation.
  = The vet treats the paw to avoid surgery.

So the original sentence is possible, but context matters.

Because behandla usually takes a direct object in Swedish.

So you say:

• behandla tassen = treat the paw
• behandla patienten = treat the patient

You do not normally need a preposition there.

English sometimes also works the same way: treat the paw, treat the wound.

Yes. In medical contexts, operera means to operate on someone/something or to perform surgery.

Examples:

• Läkaren opererar patienten. = The doctor operates on the patient.
• Hon måste opereras. = She must have surgery / be operated on.

In your sentence, it clearly has the medical meaning.

The main clause follows normal Swedish word order:

• Veterinären = subject
• behandlar = finite verb
• tassen = object

Then comes the purpose phrase:

• för att slippa operera

So the structure is:

• Subject + verb + object + purpose expression

This is a very common pattern in Swedish.