Jeg lægger låget på dåsen igen, så jeg kan gemme resten til i morgen.

Because lægge and ligge are a very common Danish pair:

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

In this sentence, jeg am actively putting the lid onto the can, so Danish uses lægger.

Compare:

• Jeg lægger låget på dåsen. = I put the lid on the can.
• Låget ligger på dåsen. = The lid is lying on the can.

So the key idea is:

• lægge takes an object: låget
• ligge describes position, not an action of placing

In Danish, the definite article is usually attached to the end of the noun.

So:

• et låg = a lid
• låget = the lid

and

• en dåse = a can / tin
• dåsen = the can

This is one of the big differences from English. Instead of saying the lid, Danish usually says lid-the.

The ending depends on the noun’s gender:

• common gender nouns usually take -en
• neuter nouns usually take -et

Here:

• låg is a neuter noun: et låg → låget
• dåse is a common gender noun: en dåse → dåsen

Because Danish, like English, usually needs the noun to be marked as definite if you mean a specific thing.

• på dåsen = on the can
• på en dåse = on a can
• på dåse by itself is not natural here

In this sentence, we are clearly talking about a specific can that already exists in the situation, so dåsen is definite.

Here på means on, but in Danish it is also the natural preposition used with a lid and a container in this kind of expression.

So:

• lægge låget på dåsen = put the lid on the can

This works much like English put the lid on the can.

You may also hear shorter versions when the container is understood:

• Jeg lægger låget på. = I’m putting the lid on.

Igen means again.

So lægger låget på dåsen igen means that the speaker is putting the lid on again, probably after having opened the can.

Its placement is natural because Danish often puts igen later in the clause, after the object or prepositional phrase.

Very literally:

• Jeg lægger låget på dåsen igen.
• I put the lid on the can again.

You might also think of it as back on again in natural English.

Så jeg kan gemme ... means so I can save/store ...

This is a very natural way in Danish to express purpose or intended result:

• Jeg lægger låget på dåsen igen, så jeg kan gemme resten til i morgen.

It focuses on what this action makes possible:
I put the lid back on, so I can save the rest for tomorrow.

A different option would be:

• ... for at gemme resten til i morgen
• ... for at kunne gemme resten til i morgen

These are also possible, but they feel a bit different:

• så jeg kan ... = so that I can ...
• for at ... = in order to ...
• for at kunne ... = in order to be able to ...

The version with så jeg kan is very idiomatic and conversational.

Because Danish modal verbs do not use at before the infinitive.

So:

• jeg kan gemme = I can save/store
• not jeg kan at gemme

This is the same pattern you get with other modal verbs:

• jeg vil spise = I want to eat
• jeg skal gå = I must / have to go
• jeg må tage det = I may / must take it

So after kan, the next verb stays in the bare infinitive: gemme.

Yes, but only in the store / keep for later sense.

Gemme often means:

• to keep
• to store
• to put away
• to save for later

In this sentence, it means keeping the remainder so it can be used another day.

That is important because English save has several meanings, and Danish does not always use gemme for all of them.

For example:

• gemme mad = save/store food for later
• spare penge = save money
• redde nogen = save someone’s life

So here gemme is the correct verb because it is about keeping the rest.

Resten means the rest or the remainder.

It works on its own as a noun, so no extra noun has to be stated.

For example:

• Jeg spiser halvdelen nu og gemmer resten.
  = I eat half now and save the rest.

In this sentence, resten refers to whatever remains of the contents of the can. Danish does not need to repeat a noun like maden or indholdet if it is already obvious from context.

And again, it is definite:

• resten = the rest

Til i morgen means for tomorrow or until tomorrow, depending on context.

In this sentence, the natural idea is:

• gemme resten til i morgen = save the rest for tomorrow

This is a very common Danish pattern:

• til senere = for later
• til i aften = for tonight
• til næste uge = for next week

Also notice that i morgen does not take an article. Danish time expressions often work without one:

• i dag = today
• i morgen = tomorrow
• i går = yesterday

Not in the same way as a word like fordi.

Here så means so, and it links the first clause with a result or purpose clause:

• Jeg lægger låget på dåsen igen, så jeg kan gemme resten til i morgen.

The word order after så is normal clause word order:

• jeg kan gemme

That is why you see jeg before kan.

This is perfectly normal and very common in Danish.

Because there are two finite clauses:

• Jeg lægger låget på dåsen igen
• så jeg kan gemme resten til i morgen

Danish uses commas between clauses, so the comma here is standard.

For learners, the easiest rule is: if you have two full clauses joined together, a comma is often needed.

Yes, you might hear that too, and it would be understandable.

But lægge is very natural here because it often describes placing something down onto a surface or object. A lid is something you physically put onto the can, so lægger works very well.

Sætte can also mean put / set / place, especially when something is placed into position, but lægge låget på is an especially common combination.

So:

• lægge låget på = very natural
• sætte låget på = also possible in many contexts

A rough guide for an English speaker would be:

• lægger: something like LEH-ger
• låget: roughly LAW-yet or LAW-get, depending on how carefully it is spoken
• dåsen: roughly DOH-sen

A few notes:

• æ in lægger is not like English a in day; it is more open, closer to e in bed, but not exactly the same.
• å in låget and dåsen is like a broad aw sound.
• In normal speech, Danish pronunciation is often softer and less clearly pronounced than the spelling suggests.

If you are learning pronunciation, these words are worth listening to from native audio, because Danish vowels and soft consonants are hard to guess from spelling alone.