Hon gav mig en hammare och en skruvmejsel, men jag behövde bara skruvmejseln.

Because mig is the object form of jag.

• jag = I (subject form)
• mig = me (object form)

In this sentence, hon is the subject — she is doing the action.
Mig is the person receiving something, so Swedish uses the object form:

• Hon gav mig en hammare. = She gave me a hammer.

You would not say Hon gav jag ..., just like you would not say She gave I ... in English.

Because both nouns are singular indefinite nouns, and both are en-words.

So:

• en hammare = a hammer
• en skruvmejsel = a screwdriver

Swedish normally repeats the article before each noun in a list like this:

• en hammare och en skruvmejsel

That is the natural equivalent of a hammer and a screwdriver.

The first time, the screwdriver is being mentioned as indefinite:

• en skruvmejsel = a screwdriver

The second time, it refers to that specific screwdriver already mentioned, so Swedish uses the definite form:

• skruvmejseln = the screwdriver

For many en-words, the definite singular is made by adding -n:

• en hammare → hammaren
• en skruvmejsel → skruvmejseln

So the sentence moves from a screwdriver to the screwdriver, just like English often does.

Because Swedish usually marks definiteness by adding an ending to the noun itself.

So:

• skruvmejseln = the screwdriver

Using den skruvmejseln is possible, but it usually adds extra emphasis, contrast, or a more specific tone, something like:

• that screwdriver
• that particular screwdriver

In this sentence, plain skruvmejseln is the normal choice.

Both are in the past tense (also called the preterite) because the sentence describes completed actions in the past.

• gav = past tense of ge (to give)
• behövde = past tense of behöva (to need)

So:

• ge → gav
• behöva → behövde

Notice that gav is irregular, while behövde is a more regular past-tense form.

Because bara means only / just, and here it is focusing on the screwdriver.

• jag behövde bara skruvmejseln = I only needed the screwdriver

Placing bara before the thing being focused on is very common in Swedish.

If you move bara, the emphasis can shift. For example:

• Jag bara behövde skruvmejseln can sound more like I just needed the screwdriver in a slightly different, more conversational sense.

So in your sentence, bara skruvmejseln clearly means only the screwdriver.

Because men is a coordinating conjunction meaning but, and what follows it is a normal main clause.

Swedish main clauses follow the V2 rule: the finite verb normally comes in the second position.

Here the subject comes first:

• jag = first position
• behövde = second position

So:

• men jag behövde bara skruvmejseln

This is normal Swedish word order.

There is no inversion here, because nothing has been placed before the subject inside that second clause except men, which does not count as the first element of the clause in the same way an adverb like sedan would.

Not always. In modern Swedish, commas are often used a bit more sparingly than in English.

Before men, a comma is often optional, especially in a short, clear sentence like this. So both of these can be acceptable:

• Hon gav mig en hammare och en skruvmejsel, men jag behövde bara skruvmejseln.
• Hon gav mig en hammare och en skruvmejsel men jag behövde bara skruvmejseln.

The comma helps mark the contrast, so many writers like to include it.

Because Swedish very often forms compound nouns as one word.

• skruv = screw
• mejsel = chisel / tool blade

Together:

• skruvmejsel = screwdriver

This is extremely common in Swedish. English often uses two words, or sometimes a hyphen, where Swedish prefers one combined word.

So a long-looking Swedish noun is often just a compound made of smaller familiar parts.

Yes, the meaning would shift a little.

• bara skruvmejseln = only the screwdriver
  → that specific screwdriver already mentioned

• bara en skruvmejsel = only a screwdriver
  → any screwdriver / a screwdriver in general

In your sentence, skruvmejseln is used because the screwdriver has already been introduced earlier. That makes the sentence more specific and natural.