Then people brought little children to Jesus for him to place his hands on them and pray for them. But the disciples rebuked them.
Jesus said, "Let the little children come to me, and do not hinder them, for the kingdom of heaven belongs to such as these." When he had placed his hands on them, he went on from there.
Matthew 19:13-15, TNIV
Many preachers I know tend to ignore these verses. Possibly that is because they do not see many resemblances between themselves - rushed off their feet, always trying to stir others into action, wanting to see God's church grow – and little children. And these really are very little. The word used - paidion - really means “infant”, but can be used of bigger dependent children. Here, they are brought to Jesus, which suggests they aren't really mobile yet. In a description of the same event, Luke uses the word brephos which clearly means “baby”.
Probably the majority of sermons I have heard on these verses (and on the parallel passages – Mark 10:13-16 and Luke 18:15-17) tend to focus not on what Jesus meant when he said this, but on what the preacher would have meant if they had said it. So they look at what babies mean to them, and from that examine what it means to receive the kingdom of heaven as a little child. (It's pretty much the same as happens when they speak about people being the salt of the earth – they don't look at what salt means in the Bible and what it would have meant to the hearers; they just think about what it means to them or used to mean to people 500 years ago. Incidentally, salt in the Bible tends to either represent the covenant or be about God bringing judgement.)
Some people do a good job at looking at the context, particularly in Mark and Luke, and get the idea that it's about holding onto God and not seeking to get into heaven on the basis of what we do. And that's certainly true, but I think we can do better.
So what did Jesus mean?
We get a very big hint because Matthew 19:13-15 isn't the first time Matthew has mentioned little children. A chapter earlier, we get this...
At that time the disciples came to Jesus and asked, "Who, then, is the greatest in the kingdom of heaven?"
He called a little child, whom he placed among them. And he said: "Truly I tell you, unless you change and become like little children, you will never enter the kingdom of heaven. Therefore, whoever takes a humble place — becoming like this child — is the greatest in the kingdom of heaven. And whoever welcomes one such child in my name welcomes me.
"If anyone causes one of these little ones—those who believe in me—to stumble, it would be better for them if a large millstone were hung around their neck and they were drowned in the depths of the sea..."
Matthew 18:1-6, TNIV
Jesus uses talking about little children as a springboard for discussing “kingdom ethics”. But what he says first is that we need to repent and become like children, and that what matters specifically is humbling oneself like the child. Now I don't tend to think of children as especially humble. I certainly wasn't especially humble as a child. But in a culture where children had a lower status, it means more. Specifically, I think what Jesus is talking about here is becoming like a child in terms of rejection of status, and rejection of trusting in ourselves.
We see that in the way that the becoming like a child stories in Matthew 19, Mark 10 and Luke 18 are all linked to the Rich Young Ruler, whose problem was that he was holding onto his wealth and his status. Jesus applies the need to become like a little child by telling the rich young man that he needed to give all his money away. That was what it meant for him.
Strikingly, and this was what got me thinking about this question originally, we see the same sort of thing in Psalm 131.
My heart is not proud, O LORD,
my eyes are not haughty;
I do not concern myself with great matters
or things too wonderful for me.
But I have stilled and quieted my soul;
like a weaned child with its mother,
like a weaned child is my soul within me.
O Israel, put your hope in the LORD
both now and forevermore.
Psalm 131, TNIV
Once again, the being like a child (this is a different word, specifically to do with being weaned) is connected to humility. But also to contentment. The weaned child is calm, and does not have any ambitions beyond being with its mother. Likewise, becoming like a child for us means rejection of ambitions, rejection of status, and simply being content to be with God. And actually, isn't that what Jesus did? He laid down all of his status, to the point of dying on a cross, and was content merely to do his Father's will.
Sadly, with the busyness of life, that is a place that many preachers find it hard to be. We have too many ambitions and plans, even if they are ambitions for God's church, and not enough contentment and becoming like little children.
I suggest reading through Psalm 131 slowly a few times, and praying through it, as a good start to a remedy.