Learn Italian: Spero ti rassicuri. - I hope it reassures you.

This is exactly what you are learning in this lesson! After verbs of hoping, believing, and fearing (like sperare, credere, and temere), you have the option to omit the conjunction che (that) when introducing a subjunctive clause. Saying Spero che ti rassicuri is equally correct, but dropping the che is extremely common and natural in modern Italian.

What is the subject of the verb rassicuri here?

The subject is an implied "it" (such as a situation, a message, or an event). Because the subject of the main clause ("I" hope) is different from the subject of the dependent clause ("it" reassures), the verb sperare triggers the subjunctive mood for rassicuri.

The verb rassicuri ends in -i. Doesn't that usually mean the subject is "you" (tu)?

In the present indicative, yes. However, in the present subjunctive, regular -are verbs have the exact same ending (-i) for the io, tu, and lui/lei/esso forms. We rely on context to know the subject is "it" rather than "you" or "I". The pronoun ti clearly tells us that "you" is the object receiving the reassurance.

Is ti functioning as a direct or an indirect object in this sentence?