Trying to explain in almost pure simple english, we have:
$$
\nabla\times\mathbf B = \mu_0\mathbf J
$$
This means the current is the source of the magnetic field. $\nabla\times$ is an operator which says how the magnetic field will behave when there is a current. Basically, this operator is known as curl operator. It basically acts on "rotating" stuffs. So, if it acts on a non-rotating stuff, it must be zero. So, the curl of a irrotational field is zero.
An easy way to visualize the operation of the curl operator when current is impressed, is to analyse the magnetic field generated by a wire. The wire is straigh up (irrotational), and so as the current. Hence because of the curl operator acting on the magnetic field, the magnetic field generated is completely rotational.
The divergent operator $\nabla\cdot$ acts on non-rotating stuffs. This means, the divergence of a field which only has rotational contributions, is zero. Hence, when we apply the divergent on both sides:
$$
\nabla\cdot\nabla\times\mathbf B = \mu_0\nabla\cdot\mathbf J = 0 \quad\Longrightarrow\quad
\nabla\cdot\mathbf J = 0
$$
This means the current must be divergence-free. Which in other words, "rotating" or "stopped". Of course, both divergent and curl operators are related to spatial variation rates. So, if the current is uniform, there will be no divergence and no curl. So, now we have the following conclusion: The current must be "rotating", or must be uniform.
However, from charge conservation we have:
$$
\nabla\cdot\mathbf J = -\frac{\partial\rho}{\partial t}
$$
This means, the field won't be uniform if there is variation of charge density (which makes intuitive sense). Also, all "rotating" contributions will go to zero, because it is no longer divergence-free. Now, all we have to think, is a situation where the magnetic field is created and we have variation of charge.
This is quite simple: Imagine a capacitor being charged. Since the charge of the plates is varying with time, then its density is also varying. Thus, the current won't be uniform. Hence the current is not divergence-free. But we saw that, if this $\nabla\times\mathbf B = \mu_0\mathbf J$ is true, then the current must be divergence-free. Contradiction. Hence, $\nabla\times\mathbf B = \mu_0\mathbf J$ is not true.
To fix this, it is necessary to add a displacement current. Maxwell saw it, and fixed it.