It is a reflexive verb, sentirsi, meaning to feel. The subject of the reflexive verb is also the object of it. Read about them here: http://italian.about.com/od/verbs/a/italian-verbs-reflexive.htm
He feels the fabric -> sente il tessuto (not reflexive)
He feels well -> He [himself] felt well -> si sente bene
Movement related verbs usually take essere/è/sono etc. For example, Io sono andata, lui è tornato.
Verbs not in this category generally use avere/avete/ha etc., such as Io ho bevuto, lei ha parlato.
This rule should work for the most part, but you'll have to just learn the exceptions by memory.
That's true, "Sentire" is defined by "to hear" and "to feel" in English, but you will differentiate them based on context. English has several verbs like this too, like "take off" or "take out". Different phrases will point to the meaning, like "senti?" = "are you listening?" or like "Mi sento bene" = "I feel good"
If anyone needs help understanding "si" and the sentence structure, you have to understand that sentirsi, which is a reflexive verb. Exactly like in French, a reflexive verb in the past tense uses the verb "to be" *essere rather than "to have" avere. So like in French, Il s'est senti bien = He felt good, in Italian it's no different. Lui si è sentito bene