Learn Spanish (Latin America): Él anhela desarrollar sus habilidades, así que estudia mucho. - He yearns to develop his skills, so he studies a lot.

Because the subject is the same for both actions! The person who yearns (anhela) is the same person who will develop (desarrollar) the skills. When the subject doesn't change after a verb of emotion or desire, we use the infinitive directly. If he wanted someone else to develop their skills, you would need the subjunctive: Él anhela que ella desarrolle sus habilidades.

Is anhelar common, or is it better to just use querer or desear?

Anhelar means to yearn or to long for. It is more formal and poetic than querer (to want) and stronger than desear (to desire). While you wouldn't use it to order a coffee, it is a great word for talking about deep goals, dreams, or ambitions like developing new skills.

Do I need a preposition like a or de between anhela and desarrollar?

Nope! Because the English translation is yearns to develop, it is very tempting to add a preposition like a or de before the infinitive. However, anhelar connects directly to the following infinitive verb without any extra words in between.

What does así que mean here, and can I use entonces instead?

Así que means so or therefore, and it connects a cause to its result (he yearns to develop his skills, so he studies). You could absolutely use entonces or por eso here instead. Así que is simply a very common, natural way to express so as a consequence in conversational Spanish.