First of all, "finito" is not one of the special verbs (modal, perception, feeling or causative) after which an infinitive can follow directly, so we need some preposition in front of the infinitive.
As the lesson's Tips and notes tell us, di can express specification. We can talk about someone having finished doing one thing or another. Specifically, we are talking about reading, so it is "di leggere".
Sorry, that is not how you say it in English. The continuous form of the infinitive has to be used in this case, so you have to say "reading" rather than "to read". That is because of the verb "to finish". With some other verbs it is OK to use "to read" e.g. "They have started to read" is OK. This probably has something to do with the fact that when you finish something it is something that you have been doing for some time, and therefore the form of the verb that follows has to show something of the continuous nature of the activity that you have finished. However when you start something, as in the second example, this does not apply and so the "to" form of the infinitive is OK (you can also say "started reading" so both forms are OK after the verb "to start").
Tips and notes of Verbs: Infinitive 1:
The Italian infinitive has many uses.
- It's the common way to turn a verb into a noun, something for which English tends to prefer the gerund
- It's used for negative or generic imperatives
- It's also used in many, very common, infinitive propositions
As a rule, when the subject of a subordinate sentence is the same as the one of the main sentence, the subordinate should be "shortened" into an infinitive proposition: but in many cases the same happens for subordinates referring to the object of the main sentence, or acting as its object.