"I feel very ill."
Translation:Me siento muy enfermo.
Here is an excellent dictionary (seriously. It´s awesome: verb conjugators, many languages, fora for usage question, etc) with the definitions of ¨sentir¨ and ¨sentirse.¨ Check out discussions on a forum for more info from native speakers.
And here's a "rule of thumb" from another source:
"The difference in using sentir and sentirse is that sentir is typically followed by a noun, while sentirse is followed by an adjective or adverb describing how a person feels."
Because sentir and sentirse are two different verbs that mean two different things. Sentir might be something like "I feel compassion for her," while sentir would be something like "I feel compassionate." One is how you feel about something else, while the other is about how you yourself are feeling.
It's not a question of using me or yo. Yo was just left off in this case. To fully explicate:
"Yo siento" and "Yo me siento" are two different things with two different meanings. When talking about how you feel, you would use "Yo me siento" (but the "yo" can be dropped off).