Jeg havde egentlig tænkt mig at blive hjemme, men tilfældigvis mødte jeg min veninde ved kiosken.

Havde tænkt mig at is a very common Danish way to say had planned/intended to.

So:

• Jeg havde tænkt mig at blive hjemme = I had planned to stay home

The form havde tænkt is the past perfect. It shows that the plan existed before the later event in the sentence, namely meeting the friend.

The mig is part of the idiomatic expression at tænke sig at + infinitive when it means to intend to do something.

Compare:

• Jeg tænker = I think
• Jeg tænkte = I thought / was thinking
• Jeg har tænkt mig at gå = I intend / am planning to go
• Jeg havde tænkt mig at gå = I had intended to go

Here egentlig means something like actually, really, or originally.

In this sentence, it adds the idea that this was the speaker’s real plan before something unexpected happened:

• Jeg havde egentlig tænkt mig at blive hjemme
  = I was actually planning to stay home

It often gives a slightly soft, conversational nuance. It does not change the basic meaning dramatically, but it makes the sentence sound more natural and nuanced.

Yes, blive often means become, but in some expressions it means stay or remain.

So:

• blive hjemme = stay home
• literally, it is closer to remain at home

This is a fixed and very common expression in Danish.

Compare:

• Jeg er hjemme = I am at home
• Jeg bliver hjemme = I am staying home

This is because Danish follows the V2 rule in main clauses: the finite verb normally comes in the second position.

When tilfældigvis is placed first, it takes the first slot, so the verb must come next:

• Tilfældigvis mødte jeg min veninde ...

Structure:

• 1st position: Tilfældigvis
• 2nd position: mødte
• then: jeg

If you start with the subject instead, you get:

• Jeg mødte tilfældigvis min veninde ved kiosken

Both are possible, but they emphasize things a bit differently. Starting with tilfældigvis highlights the accidental nature of the meeting.

Tilfældigvis means by chance, coincidentally, or accidentally in the sense of not planned.

So:

• tilfældigvis mødte jeg min veninde
  = I happened to meet my friend

It tells us that the meeting was not arranged.

Mødte is the simple past of møde.

In Danish, the simple past is very commonly used for a completed event in a past narrative:

• ... men tilfældigvis mødte jeg min veninde ...

This is natural because the whole sentence describes past events as a little story: first the plan, then the unexpected meeting.

You could use har mødt in other contexts, but here mødte fits the narrative style better.

Ven means male friend or sometimes just friend in a general sense.
Veninde specifically means female friend.

So:

• min ven = my male friend / my friend
• min veninde = my female friend

For an English speaker, an important point is that veninde does not mean girlfriend in the romantic sense. It simply means a female friend.

Min veninde means my friend, so it refers to a specific person connected to the speaker.

• min veninde = my female friend
• en veninde = a female friend

Because the speaker is talking about a particular friend she met, min is the natural choice.

Ved kiosken means by the kiosk, at the kiosk, or near the kiosk.

The preposition ved often means by, near, or at a place.

So:

• ved kiosken = at/by the kiosk

A kiosk in Danish is often a small shop, booth, or newsstand-like place, depending on context.

Also note:

• kiosk = kiosk
• kiosken = the kiosk

The -en ending makes it definite.

Yes, absolutely.

Both are grammatical:

• men tilfældigvis mødte jeg min veninde ved kiosken
• men jeg mødte tilfældigvis min veninde ved kiosken

The difference is mainly emphasis:

• tilfældigvis first = stronger focus on the accidental coincidence
• jeg first = more neutral word order

Danish often moves adverbs to the front when the speaker wants to highlight them.

Yes, it sounds very natural.

It combines several common everyday features of Danish:

• Jeg havde egentlig tænkt mig at ... = a very natural way to talk about a plan
• blive hjemme = standard expression for staying home
• tilfældigvis = common way to say by chance
• V2 word order in tilfældigvis mødte jeg ... = normal Danish grammar

So this is a good example of idiomatic, conversational Danish.