Learn Spanish (Spain): Esta música hace que me sienta mejor.

While you might occasionally hear me hace sentir, using hacer que followed by the subjunctive is the most natural and standard way to translate "makes me [do/feel something]" in Spanish. You use this structure when one subject (the music) causes a reaction in a different subject (you).

Why does hace que trigger the subjunctive (sienta)?

The verb hacer here expresses influence or causation. In Spanish, when a verb in the main clause attempts to cause, influence, or force an outcome in a subordinate clause with a different subject, that second verb must take the subjunctive mood.

Isn't sienta the él/ella/usted form? How do we know it means "I feel" here?

In the present subjunctive, the first-person (yo) and third-person (él/ella/usted) forms of a verb are completely identical. We know this refers to "I feel" rather than "he/she feels" because of the first-person reflexive pronoun me right before the verb.

Why is there an "i" in sienta? Doesn't the infinitive just have an "e" (sentirse)?

Yes, but sentirse is a stem-changing verb. Just as the e changes to ie in the present indicative (me siento), it keeps that exact same e to ie stem change in the present subjunctive for all forms except nosotros and vosotros.