How can one intuitively think about quaternions?

Here's one way. The group of unit quaternions is isomorphic to the special unitary group $\text{SU}(2)$, the group of $2 \times 2$ unitary complex matrices with determinant $1$. This group acts on $\mathbb{C}^2$ in the obvious way, and so it also acts on lines in $\mathbb{C}^2$. (These are complex lines, so they have real dimension $2$.) The space of lines in $\mathbb{C}^2$ is the complex projective line $\mathbb{CP}^1$, and it turns out there is a natural way to think about this space as a sphere - namely, the Riemann sphere. There is a beautiful projection which is pictured at the Wikipedia article which shows this; essentially one thinks of $\mathbb{CP}^1$ as $\mathbb{C}$ plus a "point at infinity" and then projects the latter onto the former in a way which misses one point.

So $\text{SU}(2)$ naturally acts on a sphere, and as it turns out it naturally acts by rotations. This describes the famous 2-to-1 map $\text{SU}(2) \to \text{SO}(3)$ which allows quaternions to describe 3D rotations.

I find the conversion between quaternions and the axis-angle representation quite instructive.

In the axis-angle representation, you describe a rotation by specifying the axis of rotation as a unit vector $\vec\omega$ and an angle $\theta$ about which to rotate around this axis. An interesting fact is that any possible rotation can be described in this way.

The corresponding quaternion is given simply by $\left(\cos(\theta/2), \vec\omega\sin(\theta/2)\right)$. Here the notation $(a, \vec v)$, where $a$ is a scalar and $\vec v$ a real vector, denotes the quaternion $a + v_xi + v_yj + v_zk$, or $(a,v_x,v_y,v_z)$.

Here is the intuitive interpretation of this. Given a particular rotation axis $\omega$, if you restrict the 4D quaternion space to the 2D plane containing $(1,0,0,0)$ and $(0,\omega_x,\omega_y,\omega_z)$, the unit quaternions representing all possible rotations about the axis $\vec \omega$ form the unit circle in that plane. A rotation of $\theta$ about the axis $\vec \omega$ is the point at an angle $\theta/2$ from $(1,0,0,0)$ on that circle. For example, not rotating at all is $(1,0,0,0)$, rotating 180° is $(0,\omega_x,\omega_y,\omega_z)$, and rotating 360° is $(-1,0,0,0)$, which is the same as not rotating at all (see final paragraph).

Multiplying two quaternions is unintuitive, but I'm not bothered by that, because the composition of two rotations in real life is quite unintuitive in the first place.

(Note that the quaternions are a "double cover" of the space of rotations, in that any rotation actually has two quaternions, say $q$ and $-q$, that represent it: $\theta$ and $\theta + 2\pi$ are the same angle, but $\theta/2$ and $(\theta+2\pi)/2$ are not. This is the only "glitch" in the quaternion representation of rotations.)

I highly recommend that your read the presentation in Conway and Smith: On quaternions and octonions: Their geometry, arithmetic, and symmetry. Here's an excerpt from John Baez's very informative review:

It follows that the quaternions of norm 1 form a group under multiplication. This group is usually called SU(2), because people think of its elements as 2 × 2 unitary matrices with determinant 1. However, the quaternionic viewpoint is better adapted to seeing how this group describes rotations in 3 and 4 dimensions. The unit quaternions act via conjugation as rotations of the 3d space of "pure imaginary" quaternions, namely those with Re(q) = 0. This gives a homomorphism from SU(2) onto the 3d rotation group SO(3). The kernel of this homomorphism is {±1}, so we see SU(2) is a double cover of SO(3). The unit quaternions also act via left and right multiplication as rotations of the 4d space of all quaternions. This gives a homomorphism from SU(2) × SU(2) onto the 4d rotation group SO(4). The kernel of this homomorphism is {±(1, 1)}, so we see SU(2) × SU(2) is a double cover of SO(4).

These facts are incredibly important throughout mathematics and physics. With their help, Conway and Smith classify the finite subgroups of the 3d rotation group SO(3), its double cover SU(2), the 3d rotation/reflection group O(3), and the 4d rotation group SO(4). These classifications are all in principle "well known'. However, they seem hard to find in one place, so Conway and Smith's elegant treatment is very helpful

See also the Wikipedia article Quaternions and spatial rotation.