First, are these two expressions "I feel like to .." and "I feel like ..ing .." correct?
So, What's the meaning of them?
And what's the difference between them?
"I feel like to .."
isn't something I've heard said, I don't think it's generally accepted.
"I feel like ___ing"
is correct and used to express desire.
"I feel like eating" = "I want to eat"
"I feel like swimming" = "I want to swim"
In English it's common for someone to ask "What do you feel like?" or "What do you feel like doing?" (or eating, etc). This is the general response you'd use to those questions.