Intuition behind cohomology operations

You could start with Steenrod's classic article "Cohomology Operations, and Obstructions to Extending Continuous Functions* which was republished after his death. And then do a web search on the topic.

An intuition behind the cup product in cohomology is that is the map induced by the diagonal maps $ \Delta_X \colon X \to X \times X$. These maps are natural with respect to maps $X \to Y$ and the corresponding $X \times X \to Y \times Y$, hence this product is functorial.

Now consider $X^p$, the cartesian product of $X$ with itself $p$ times, where $p$ is a prime. There is a map

$$ T_X \colon X^p \to X^p$$

$$ (x_1,\ldots,x_p) \mapsto (x_2,\ldots,x_p,x_1) $$

which is also natural with respect to maps $X \to Y$. The corresponding maps $X^p \to Y^p$ would give you a commutative square. The existence of the Steenrod powers, which are cohomology operations for cohomology with coefficients in $\mathbb{Z}/p$, comes from these maps.

This is just an intuition, the real details are more intricate and technical and use the smash product instead of the Cartesian product. But I believe it makes more sense when you think about it in terms of Eilenberg-MacLane spaces.

There is a natural bijection

$$ H^n(X;\mathbb{Z}/p) \cong [X,K(\mathbb{Z}/p,n)] $$

where the right side denotes homotopy classes of maps $X \to K(\mathbb{Z}/p,n)$, then you can think of cohomology operations (for cohomology with coefficients in $\mathbb{Z}/p$) in terms of maps $$K(\mathbb{Z}/p,n) \to K(\mathbb{Z}/p,m)$$

Such a map would give you a way of getting an element of the $m$th cohomology of $X$ from an element of the $n$th cohomology of $X$ by composition. But we have

$$ [ K(\mathbb{Z}/p,n) , K(\mathbb{Z}/p,m) ] \cong H^m(K(\mathbb{Z}/p,n);\mathbb{Z}/p) $$

and so we have a cohomology operation for each element in these groups. In fact the cohomology rings of $K(\mathbb{Z}/p,n)$ were computed by Cartan and Serre and a multiplicative basis is given by the Steenrod powers and the Bockstein.