I don't think there is an "intuitive" explanation to the effect. As an ex-physicist I don't find rotational movement intuitive, maybe evolutionary we don't encounter them that often to develop senses around them. So, the answer really is in equations and breaking the symmetry of sorts: the inertia moments are ordered, so the one in the middle is different than the ones at the edges.

One thing to note is that people mostly pay attention to the case where the main rotation is around the second moment of inertia, where the seemingly sudden and violent change in direction is startling to an observer. However, in order to grasp the effect one must understand that regardless of initial state rotation direction the rotation along the second moment is always periodic, i.e. it always changes the direction! It's just sometimes it is not noticeable.

Here's the details. The main set of equation where we start is in Wikipedia: $$I_1\dot\omega_1=(I_{3}-I_{2})\omega_{3}\omega_{2}$$ $$I_2\dot\omega_2=(I_{1}-I_{3})\omega_{1}\omega_{3}$$ $$I_3\dot\omega_3=(I_{2}-I_{1})\omega_{2}\omega_{1}$$ Where $I_i$ are moments of inertia and $\omega_i$ are rotation velocities along these moments.

Now, let's take one more derivative with respect to time: $$I_1\ddot\omega_1=(I_{2}-I_{3})\left(\frac{I_1-I_2}{I_3}\omega_2^2-\frac{I_1-I_3}{I_2}\omega_3^2\right)\omega_1$$ $$I_2\ddot\omega_2=-(I_{1}-I_{3})\left(\frac{I_2-I_3}{I_1}\omega_1^2+\frac{I_1-I_2}{I_3}\omega_3^2\right)\omega_2$$ $$I_1\ddot\omega_3=(I_{1}-I_{2})\left(-\frac{I_1-I_3}{I_2}\omega_2^2+\frac{I_2-I_3}{I_1}\omega_1^2\right)\omega_3$$

I re-arranged the terms in such a way that all fractions and subtractions in parentheses are positive, so that it becomes very evident that only the rotation velocity along the second moment $\omega_2$ is always periodic: $\ddot\omega\sim-\omega$. This velocity always oscillates. This is the most important aspect of the rotational movement.

It is only when you start the rotation with a significant velocity $\omega_2(0)\gg 0$ the effect is so spectacular, where it violently rotates the body and changes the direction, but it is always there!

You can see that the dynamics of other angular velocities $\omega_1,\omega_2$ can vary depending on what's going on with other velocities which is determined by the initial speeds. Sometimes you have $\ddot\omega_i\sim\omega_i$ which corresponds to either exponential decay $\omega_i(t)\sim e^{-t}$ or explosion $\omega_i(t)\sim e^{t}$. At other times it becomes periodic too $\ddot\omega_i\sim-\omega_i$.

So, when you start with $\omega_2\gg 0$ it is more likely that other two rotations $\omega_1,\omega_3$ are likely to go through this changes of dynamics that I described above because $\omega_2$ will be smaller or larger in relation to other rotations, changing the signs of $\left(-\frac{I_1-I_3}{I_2}\omega_2^2+\frac{I_2-I_3}{I_1}\omega_1^2\right)$, for example, which will be changing the dynamics of the $\omega_3$ in this case. All while $\omega_2$ is always doing its periodic dynamics, albeit with always changing periodicity.

When $\omega_2(0)\approx 0$ then it's likely that the sign and the amplitude in $\left(-\frac{I_1-I_3}{I_2}\omega_2^2+\frac{I_2-I_3}{I_1}\omega_1^2\right)$ term and its equivalent for $\omega_1$ stable, which makes overall rotation of the body rather stable too.

I don't think there is an "intuitive" explanation to the effect. As an ex-physicist I don't find rotational movement intuitive, maybe evolutionary we don't encounter them that often to develop senses around them. So, the answer really is in equations and breaking the symmetry of sorts: the magnitudes of inertia moments are ordered, so the rotation around one in the middle is different than around the other ones.

One thing to note is that people mostly pay attention to the case where the main rotation is around the second principal moment of inertia, where the seemingly sudden and violent change in direction is startling to an observer. However, in order to grasp the effect one must understand that regardless of initial state rotation direction the rotation along the second moment is always periodic, i.e. it always changes the direction! It's just sometimes it is not noticeable.

Here's the details. The main set of equation where we start is in Wikipedia: $$I_1\dot\omega_1=(I_{3}-I_{2})\omega_{3}\omega_{2}$$ $$I_2\dot\omega_2=(I_{1}-I_{3})\omega_{1}\omega_{3}$$ $$I_3\dot\omega_3=(I_{2}-I_{1})\omega_{2}\omega_{1}$$ Where $I_i$ are moments of inertia and $\omega_i$ are rotation velocities along these moments. so the overall rotation of the body can be represented by rotations around these three principal moments.

Now, let's take one more derivative with respect to time: $$I_1\ddot\omega_1=(I_{2}-I_{3})\left(\frac{I_1-I_2}{I_3}\omega_2^2-\frac{I_1-I_3}{I_2}\omega_3^2\right)\omega_1$$ $$I_2\ddot\omega_2=-(I_{1}-I_{3})\left(\frac{I_2-I_3}{I_1}\omega_1^2+\frac{I_1-I_2}{I_3}\omega_3^2\right)\omega_2$$ $$I_1\ddot\omega_3=(I_{1}-I_{2})\left(-\frac{I_1-I_3}{I_2}\omega_2^2+\frac{I_2-I_3}{I_1}\omega_1^2\right)\omega_3$$

I re-arranged the terms in such a way that all fractions and subtractions in parentheses are positive, so that it becomes very evident that only the rotation velocity along the second moment $\omega_2$ is always periodic: $\ddot\omega\sim-\omega$. This velocity always oscillates. This is the most important aspect of the rotational movement.

It is only when you start the rotation with a significant velocity $\omega_2(0)\gg 0$ the effect is so spectacular, where it violently rotates the body and changes the direction, but it is always there!

You can see that the dynamics of other angular velocities $\omega_1,\omega_2$ can vary depending on what's going on with other velocities which is determined by the initial speeds. Sometimes you have $\ddot\omega_i\sim\omega_i$ which corresponds to either exponential decay $\omega_i(t)\sim e^{-t}$ or explosion $\omega_i(t)\sim e^{t}$. At other times it becomes periodic too $\ddot\omega_i\sim-\omega_i$.

So, when you start with $\omega_2\gg 0$ it is more likely that other two rotations $\omega_1,\omega_3$ are likely to go through this changes of dynamics that I described above because $\omega_2$ will be smaller or larger in relation to other rotations, changing the signs of $\left(-\frac{I_1-I_3}{I_2}\omega_2^2+\frac{I_2-I_3}{I_1}\omega_1^2\right)$, for example, which will be changing the dynamics of the $\omega_3$ in this case. All while $\omega_2$ is always doing its periodic dynamics, albeit with always changing periodicity.

When $\omega_2(0)\approx 0$ then it's likely that the sign and the amplitude in $\left(-\frac{I_1-I_3}{I_2}\omega_2^2+\frac{I_2-I_3}{I_1}\omega_1^2\right)$ term and its equivalent for $\omega_1$ stable, which makes overall rotation of the body rather stable too.