This is a very interesting question. My first instinct is to say that they are, by definition, *sharp* changes in the form of the distribution of probability of microstates, that occur even for *small* changes in the macrostate. So, there are plenty of examples of physical systems where changes happen gradually. Phase transitions are the ones where they don't, **by definition**.

## The Ising  example
I'm guessing this answer won't satisfy you, and it doesn't satisfy me either. So, let's have a look at the simplest example I can think of, the all-to-all Ising model without an external field. Here, every spin $i$ has a lower energy if it's aligned to average field of the others, and a higher one if it's anti-parallel to it.
$$ H_i = -Jm\sigma_i $$
where $\sigma = \pm 1$ is the spin direction, $J$ is a positive constant, and $m$ is the average field that the spin "feels". Now, the point is that this $m$ is itself *generated by the other spins*:
$ m = \sum_j \sigma_j.$
You can see, then, that there is a non-linear *self* interaction of the group of spins. Without going into the details (you probably know them  already), it turns out that there's an equation for $m$:
$$ m = \tanh(Jm).$$
It's easy to see that this has 0 as a trivial solution; however, it also has two additional solutions (as you can see by plotting both sides of the equation) when $J>1$, and these happen to be stable. This shows why there is a phase transition in this system. For $J\leq 1$, there is one solution. For $J>1$, there are two. There is no such a thing as "one solution and a half", of course, so this is necessarily a discontinuous difference.
So my second answer is: it descends from **the weirdness of certain nonlinear equations** that govern the system. This is also related to the mathematical concept of [bifurcation][1].

Having a look at [Landau theory][2] may help you understand how this is not limited to the Ising model.


## Symmetries

Another way of putting it is connected with the concept of symmetry. The disordered phase has a higher degree of symmetry than a state of the ordered phase: an Ising model with zero magnetisation is invariant under up/down flipping of the whole system; a non-synchronised [Kuramoto][3] model has undefined average phase of its oscillators, which gives it $U(1)$ symmetry.

However, once the oscillators synchronise, they will have a given global phase. When the spins align, they may randomly align in the "up" or the "down" direction, but they have to *collectively choose* one. This is referred to as [spontaneous symmetry breaking][4]. As far as I know, **symmetries can't be half-broken**.

## Gas/liquid/solid

I don't know enough about these transitions to tell you about the physical details, but my guess is that something analogous to what I described above also happens in this case. However, these are **first order** transition, which follow different formalisms, and don't obey Landau theory. Some interesting points about those are made in [this answer][5] to a different question.

I hope this gives you some intuition.

  [1]: https://en.wikipedia.org/wiki/Bifurcation_theory
  [2]: https://en.wikipedia.org/wiki/Landau_theory
  [3]: https://en.wikipedia.org/wiki/Kuramoto_model
  [4]: https://en.wikipedia.org/wiki/Spontaneous_symmetry_breaking
  [5]: https://physics.stackexchange.com/q/314134