Learn Dutch: Wij besteden aandacht aan de ademhaling. - We pay attention to the breathing.

In English you 'pay' attention, but the fixed Dutch expression is aandacht besteden aan (literally: to spend attention on). You should never use the Dutch verb for 'to pay' (betalen) in this context.

Does besteden always take the preposition aan?

Yes, when you want to specify what you are spending something on, you use the fixed combination besteden aan. This applies whether you are spending time, money, or attention (aandacht).

How is the noun ademhaling formed?

It is built from the noun adem (breath) and the verb halen (to fetch or pull). Together they make the verb ademhalen (to breathe). Adding the suffix -ing turns it into the noun for the continuous process: breathing or respiration.

Why does the sentence use de ademhaling instead of a possessive pronoun like onze ademhaling (our breathing)?

You can absolutely say onze ademhaling. However, Dutch frequently uses the definite article (de or het) for body parts and bodily functions when it is already obvious whose body is involved. Since the subject is wij (we), the 'our' is naturally implied.