They don't. You don't know if they do. The majority of people who voted for Yanukovich are the people in the east, pretty far from Kiev. It is THOSE people, in Crimea, and the rest of the east, who are not acknowledging new Kiev revolutionaries. It is the act of allowing this revolution to transpire, as opposed to a democratic process, that is really inviting war and division.
If they waited a year and then elected a pro-Western president, none of this would have happened.
The people are the state. Whatever the majority of the people wants is what the government has to do. So IMHO the right thing to do in such a situation is to conduct an out of order referendum to establish what the majority wants (and I would argue it should be a clear majority like 2:1 to justify exceptional measures) and if the majority agrees that they want to replace their elected president immediately then the president has to resign immediately. If he does not follow the will of the majority, the people have the right to remove him with force. I am of course aware that it might be quiet hard to conduct a representative referendum about a highly controversial topic and probably even without support of the current government.
If they waited a year and then elected a pro-Western president, none of this would have happened.