This is a difficult question because the Bible does not tell us directly why Jesus did this. In several passages, Jesus warns people "not to tell" anyone that he is the messiah or that he had just healed them or both. Jesus also tells demons to be quiet when they know and proclaim who he is.

It is easy to see why Jesus wouldn't want demons telling everyone who he was — they were unreliable evil witnesses. However, why wouldn't Jesus allow people to tell what he had done for them and who he was?

I think the Bible gives us a few different answers to this question. First, in John 2:3-4, with the account of the wedding at Cana:

When the wine was gone, Jesus' mother said to him, "They have no more wine."
"Dear woman, why do you involve me?" Jesus replied, "My time has not yet come."

Jesus said that his time had not yet come, and I think this is one of Jesus' concerns when he tells people not to spread the word, and to keep his identity as messiah a secret. Without getting into exactly what Jesus might have meant by this, I think it is safe to say that the first explanation that we are given for this phenomenon is that Jesus was concerned with timing, specifically that events would not move too rapidly as a result of his widespread fame.

A second reason why Jesus might have been concerned about people telling what they knew about him was that he would be unable to travel freely in the towns, and thus it would be more difficult to reach people who couldn't come to the countryside to hear him preach. In Mark 1:40-45 we have an account of Jesus healing a leper and instructing him not to tell anyone. The leper, however was unable to contain himself.

Mark 1:45 —

Instead he went out and began to talk freely, spreading the news. As a result, Jesus could no longer enter a town openly but stayed outside in lonely places. Yet the people still came to him from everywhere.

We see here that as a result of the leper telling everyone, Jesus' ministry was hampered in that he could no longer travel openly in the towns, forcing him to stay in the "lonely places" or the countryside.

A third reason that the gospels imply that Jesus did not want people reporting that he was the messiah probably had to do with the fact that first century Jews has many misconceptions about who the messiah was. Matthew 12:15-21 —

Aware of this, Jesus withdrew from that place. Many followed him, and he healed all their sick, warning them not to tell who he was. This was to fulfill what was spoken through the prophet Isaiah:

"Here is my servant whom I have chosen,
the one I love, in whom I delight;
I will put my Spirit on him,
and he will proclaim justice to the nations.
He will not quarrel or cry out;
no one will hear his voice in the streets.
A bruised reed he will not break,
and a smoldering wick he will not snuff out,
till he leads justice to victory.
In his name the nations will put their hope."

This passage indicates that Jesus warned people not to tell who he was because Jesus wanted to fulfill Isaiah's prophecies regarding the suffering servant. It seems that Jesus thought that if his fame spread too much then this prophecy would be less likely to be fulfilled. This could be because too many of his followers would want to make him into a political messiah and military leader instead of the suffering messiah of the prophecies of Isaiah.