It is clearer as a question. Kids wants to stay home from school and mother asks "Are you really sick or do you just not want to go to school?" She is not questioning whether the degree illness, but the reality of it. When you say "I am really sick" to express the degree jof sickness what you are really saying is that you are sick enough to know that it is real.
I think "sentir" is a transitive verb, i.e. it always takes on a direct object, unlike "feel" in English which can be transitive or intransitive. To translate "I feel" (intransitive), you have to make it reflexive in Spanish: sentirse ("to feel oneself", as it were). Perhaps it helps to think of "Me siento enfermo" as "I feel myself (being) ill".
Can someone explain the rule of yo vs me? Can you use both 'yo me' as suggested? I do see a reference in my dictionary to me being reflexive but in that context it uses myself not me. Just found a good link http://www.studyspanish.com/lessons/reflexive1.htm
litteral translation yo=I/me=myself (reflective pronoun)/siento= feel/ enfermo= sick; I myself feel sick; Sentir is reflective in Spanish but no longer reflective in modern English. So the verb being used here is 'sentirse' and not 'sentir'. Edit: 'me' also means 'me' and 'to me, or for me', but not here in this sentence.