This is probably because Lennon, a student of human nature, knew that government may be one thing we can't do without. But let's do what he didn't suggest and imagine, for a moment, a world with no government. Let's imagine that in this world with no government, I'm typing this article for you on my MacBook right now. And let's imagine that there's a very large man--we'll call him Biff--who doesn't especially like my writing, so he walks in, throws the MacBook on the floor, stomps it into little pieces, and leaves. And before leaving, Biff tells me that if I write anything else he doesn't like, he'll do to me what he did to my MacBook.
Well, in doing that he just established something very much like his own government. It is now, as a matter of practice, against Biff's law for me to write things that Biff doesn't like. The penalty is severe, enforcement fairly certain (at least within this jurisdiction). And who's going to stop him? Certainly not me; I'm smaller and less violent than he is.
But Biff isn't really the biggest problem in this no-government world anyway. The real problem is a really greedy, heavily armed guy--we'll call him Frank--who has learned that if he steals money and then hires enough muscle with his ill-gotten gains, he can demand goods and services from every business in town, take anything he wants, and make almost anybody do whatever he says. And since there's no authority higher than Frank that can make him stop what he's doing, this jerk just literally created his own government--what political theorists refer to as a despotism, a government ruled by a despot (which is essentially just another word for a tyrant).
Some governments aren't much different from the despotism I just described. In North Korea, for example, Kim Jong-il technically inherited his army instead of hiring it--but the principle is the same. What Kim Jong-il wants, Kim Jong-il gets. It's the same system Frank used, but on a larger scale.
If we don't want Frank (or Kim Jong-il) in charge, we have to all get together and agree to do something to prevent them from taking over. And that agreement itself is a government.
In other words: The reason we need governments is to protect us from other, worse power structures that would otherwise form in our midst and deprive us of our rights. As Thomas Jefferson put it in the Declaration of Independence (emphasis mine):
We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable rights, that among these are life, liberty and the pursuit of happiness. That to secure these rights, governments are instituted among men, deriving their just powers from the consent of the governed, that whenever any form of government becomes destructive of these ends, it is the right of the people to alter or to abolish it, and to institute new government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their safety and happiness.See also: Why Laws Exist