Learn Russian: Эти государства оказывают сильное влияние на развитие общества. - These states exert a strong influence on the development of society.

This is a formal analytical predicate. In formal, journalistic, or academic Russian, it is very common to replace a simple verb like влиять (to influence) with a noun-verb collocation. Using the light verb оказывать (to exert) plus the noun влияние sounds more official. It also makes it much easier to attach adjectives, as seen here with сильное (strong).

Do we always use the preposition на after влияние?

Yes, when you want to specify what is being influenced. Just like the base verb влиять, the noun влияние requires the preposition на followed by the accusative case to indicate the target of the influence.

Why is развитие in the accusative case after на, rather than the prepositional case?

The preposition на takes the accusative case when it indicates the direction or target of an action, including abstract targets like influence. It only takes the prepositional case when indicating a static location. Since развитие is a neuter noun ending in -ие, its accusative form is identical to its nominative form.

Can I translate 'to exert' with a different verb here, like делать?