Tom offered _____ us to the railway station
The given answer is "to drive".
After searching in a dictionary, I understood that "drive" is used as a noun, and "to" might be used as a preposition.
Why can't we use "drive" instead of "to drive"?
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You have misunderstood.
Offer can function as a catenative verb—that is, verbs that can be followed directly by another verb. Often, the second verb is a gerund. In other cases, the verb is an infinitive.
That is the case here: Tom offered [verb] + to drive [infinitive]
Wiktionary provides these similar examples (and many more):
He agreed to work on Saturday.
I promise to tell the truth.
You don’t deserve to be treated like that.
In addition to Jeffrey's answer I'd just like to mention that this is a pattern you might recognize from some pretty simple English phrases. Take, for example, someone that is tired and wants to leave a party:
I want to leave
Now consider your question, slightly changed:
Why can't we use "leave" instead of "to leave"?
You probably recognize that "I want leave" sounds really strange. Jeffrey's response answers why:
want is also a catenative verb.