The typical putz-in-the-street hasn't a clue what "impeachment" is for.
So, get out the old Civics 101 books.
The president and legislators are, in a very real though limited sense, above the law. The reasons are based in history. The guys who invented the structure of US government knew their Herodotus, their Plutarch, their Tacitus, their Blackwell, and, after 1776, their Gibbon. The historical weaknesses of representative government were well known.
One such weakness was that one branch could use the law to harrass another branch to the point of nonfunctionality. If the president could send the police to detain the members of congress for some plausible, even if false, charge - crossing the yellow line while on the way to the Capitol, say - then he could prevent one party or the other from turning up to vote on a scheduled measure, or even prevent congress as a whole from achieving a quorum. Similarly, congress could cause the president to miss, say, an important treaty signing by sending the police to arrest him for spitting on the sidewalk. The police, after investigation, would doubtless release him without charge once they investigated and found that he had not spit on the sidewalk, but by then the damage would have been done.
To prevent this, neither the president, not congress while in its chambers or while on the way to or from the chambers, can be apprehended and detained by the police. That is, they're arrest-proof.
That raises another possibility. Suppose that the president walks into congress while in session, pulls out a duelling pistol, and shoots the speaker of the house dead. A farfetched scenario, but not so farfetched then as it would be today. What to do? He can't be arrested. At least, not while he's president. The solution is to remove him from office. After that, the law can take its normal course.
The removal from office is impeachment. Impeachment is not, in itself, a law-enforcement measure. Congress has no law-enforcement powers, and as the courts have reaffirmed, it can't; the separation-of-powers violation should congress play policeman or judge is obvious enough.
Of course the impeachment function was abused almost immediately (well, certainly by Jefferson's term), with the attempted impeachement of one of the Federalists on the Supreme Court, Samuel Chase. I've gone through that story before on these very pages so won't do it again. Chase hadn't taken a shot at anybody; the Democratic-Republicans just wanted to get rid of a Federalist so that Jefferson could appoint a good Democratic-Republican to the bench in his place. The impeachment failed dismally. Sixty-odd years later, an impeachment was launched against Andrew Johnson when congress tried to assert control over the president's office with the Tenure of Office Act. Johnson wanted to fire Stanton, the secretary of war, and congress tried to prevent him from doing so. Stanton actually barricaded himself into his office for a week or so. That impeachment failed dismally also. The Tenure of Office Act was later judged by the Supremes to be unconstitutional, which was handy.
Attempts to use impeachment for mere political gain are an abuse of constitutional government, and we're fortunate that nearly all such attempts have failed.