Jeg lægger hammeren tilbage i skuffen, efter at jeg har sat det sidste søm i væggen.

This is a very common Danish learner question.

• lægge = to lay / put something somewhere
• ligge = to lie / be lying

So:

• Jeg lægger hammeren i skuffen = I put/lay the hammer in the drawer.
• Hammeren ligger i skuffen = The hammer is lying in the drawer.

In your sentence, jeg is actively moving hammeren, so Danish uses lægger, not ligger.

Danish often chooses different verbs depending on the position an object ends up in.

Very roughly:

• lægge = put something so it lies flat / is laid down
• stille = put something so it stands upright
• sætte = put something so it sits / is attached / is set in place

A hammer in a drawer is naturally thought of as something you lay down, so lægger is a good choice.

So lægger hammeren i skuffen feels more natural than using one all-purpose verb for put.

Tilbage means back.

So:

• Jeg lægger hammeren i skuffen = I put the hammer in the drawer.
• Jeg lægger hammeren tilbage i skuffen = I put the hammer back in the drawer.

It tells you that the drawer is the hammer’s earlier or usual place.

Because they are in the definite singular form: the hammer, the drawer.

In Danish, the definite article is often added as an ending:

• en hammer = a hammer

• hammeren = the hammer

• en skuffe = a drawer
• skuffen = the drawer

So Danish often uses an ending where English uses the.

Because Danish handles definiteness differently when an adjective comes before the noun.

Here:

• søm is a neuter noun: et søm
• so the definite article before it is det
• sidste is the adjective: last / final

That gives:

• det sidste søm = the last nail

In Danish, when you have definite article + adjective + noun, the noun usually does not take the definite ending.

So:

• sømmet = the nail
• det sidste søm = the last nail

Not det sidste sømmet.

Also, sidste has -e because it is the adjective form used before a noun in this kind of phrase.

Har sat is the present perfect of sætte.

• at sætte = to set / put
• har sat = have set / have put

In this sentence, it shows that the nail action is completed before the hammer is put away:

• first: jeg har sat det sidste søm i væggen
• then: jeg lægger hammeren tilbage i skuffen

So the perfect helps mark the nail as already dealt with before the next action.

English can do something similar with after I have put/set the last nail in the wall, though English often also just says after I put.

Because Danish uses different placement verbs for different kinds of results.

• lægge hammeren i skuffen: you lay the hammer down
• sætte et søm i væggen: you set/put a nail into the wall

A nail is not just laid somewhere; it is put into the wall and ends up fixed there. That makes sætte a reasonable choice.

You should also know that in everyday Danish, slå et søm i væggen is very common too, especially if you want to emphasize the hammering action itself.

Danish i can often cover both in and into, and the verb tells you whether movement is involved.

So:

• lægger hammeren i skuffen = put the hammer in/into the drawer
• sat et søm i væggen = put/set a nail in/into the wall

If you want to emphasize movement inward, Danish can use ind i:

• lægger hammeren ind i skuffen

But in many normal sentences, simple i is enough.

Because the nail goes into the wall, not just on the wall.

• i væggen = in / into the wall
• på væggen = on the wall

A picture can hang på væggen, but a nail is typically i væggen.

Efter at means after and introduces a subordinate clause.

That is why you get:

• efter at jeg har sat det sidste søm i væggen

In this subordinate clause, the subject comes before the finite verb:

• jeg har sat

That is normal Danish subordinate-clause word order.

If you move that whole clause to the front, the main clause changes word order because of Danish V2 word order:

• Efter at jeg har sat det sidste søm i væggen, lægger jeg hammeren tilbage i skuffen.

Notice that it becomes lægger jeg, not jeg lægger, because the sentence now starts with the subordinate clause.

In modern Danish, that comma is often optional, depending on which comma style is being followed.

So both of these may be seen:

• Jeg lægger hammeren tilbage i skuffen, efter at jeg har sat det sidste søm i væggen.
• Jeg lægger hammeren tilbage i skuffen efter at jeg har sat det sidste søm i væggen.

Many learners are taught to include the comma because it makes the structure easier to see, but you will encounter both versions.