You’re thinking ‘something that is actively performed on’, i.e. an object. It is true, that if a verb takes an object and if that object is usually inanimate by nature then this object will typically be in accusative case in German:
Er spielt die Rolle sehr überzeugend.
Verbs that have an accusative object like in the above example are called transitive. The verb fahren can also be transitive:
Er fährt ein blaues Fahrrad zur Schule.
In English, a possible translation would be:
He rides a blue bike to school.
However, fahren can be used in many different ways. The above emphasises the act of riding the bike and the destination (the school) is less relevant. Another way to use fahren is intransitively. In this case, there is no direct object and the main meaning shifts from steering/driving a vehicle to using a vehicle to get from A to B. A more reduced version but still perfectly valid version of your example sentence would be:
Er fährt zur Schule.
These in themselves are complete sentences using the intransitive meaning of fahren. As with most verbs, they can be expanded with various adverbials one of which designating the tool—here, the bike. When describing a tool with an adverbial, the usual way to do it is in a mit expression. Mit as a preposition governs dative, thus in your example sentence dative is the only possible choice:
Er fährt mit einem blauen Fahrrad zur Schule.
Compared to the above, the focus here is centred more on where he is going with the additional information of how being optional.
Even without this in-depth analysis, the only possible choice given the small blank you have is blauen anyway: the indefinite article einem already makes it clear that the dative form is required.