Jeg åbner appen for at betale regningen.

Åbner is the present tense of åbne. Most Danish verbs make the present tense by adding -r to the infinitive: åbne → åbner, betale → betaler.

Danish does not normally use a separate everyday form like English am opening. So Jeg åbner appen can mean both I open the app and I am opening the app, depending on context.

The basic order here is:

subject + finite verb + object

So:

Jeg = subject
åbner = finite verb
appen = object

This is the normal pattern in a simple Danish main clause. More generally, Danish is a verb-second language, which means the finite verb usually comes in the second position in the clause. For example:

Nu åbner jeg appen.
Here Nu comes first, so the verb åbner still stays in second position, and jeg moves after it.

Because Danish usually puts the definite article at the end of the noun.

So:

en app = an app
appen = the app

en regning = a bill / an invoice
regningen = the bill

Unlike English, Danish often does not use a separate word like the before the noun. Instead, it adds a definite ending to the noun itself.

Because app and regning are common-gender nouns in Danish.

Danish nouns are mainly divided into two genders:

common gender: takes en
neuter: takes et

So:

en app → appen
en regning → regningen

If a noun were neuter, you would usually get:

et hus → huset

Unfortunately, gender usually has to be learned with each noun, so it is best to memorize nouns together with their article: en app, en regning.

For at introduces a purpose. It means something like:

to
in order to
so as to

So:

for at betale regningen = to pay the bill / in order to pay the bill

In this sentence, it explains why the speaker opens the app.

Also, at is the infinitive marker, like English to before a verb:
at betale = to pay

No, that is not natural Danish.

If you want to express purpose after a main clause, Danish normally uses for at:

Jeg åbner appen for at betale regningen.

Without for, the sentence does not work in the same way.

A different natural way to say it would be:

Jeg åbner appen, så jeg kan betale regningen.
That means I open the app so that I can pay the bill.

Because for at betale regningen is an infinitive phrase, not a full clause.

The subject is understood automatically: it is the same as the subject of the main clause, which is jeg.

So the sentence means:

I open the app [in order for me] to pay the bill.

English works similarly:

I open the app to pay the bill.

You do not normally repeat the subject there either.

Because with betale, Danish often uses a direct object when talking about a bill, a price, rent, an amount of money, and similar things.

So these are natural:

betale regningen = pay the bill
betale huslejen = pay the rent
betale 100 kroner = pay 100 kroner

Betale for is used in other situations, especially when you mean pay for something more generally:

betale for maden = pay for the food
betale for en billet = pay for a ticket

So in this sentence, betale regningen is the most natural choice.

Because the speaker is talking about a specific bill or invoice, not just any bill in general.

So:

en regning = a bill
regningen = the bill

In this context, there is clearly one particular bill the speaker wants to pay, so Danish uses the definite form regningen.

Å is a Danish vowel that does not match English exactly, but a useful approximation is the vowel in off or British bought.

So åbner starts with a sound roughly like aw-.

It is not like English oh. A rough learner-friendly approximation is:

åbner ≈ AWB-ner

That is not perfect Danish pronunciation, but it points you in the right direction.