I am not familiar with Clean; this answer addresses Lazy Evaluation in general.
Lazy evaluation happens at runtime.
The most fundamental form of lazy evaluation is leftmost reduction in lambda calculus. I will not burden this answer with a full introduction to lambda calculus. The intuition is that terms in lambda calculus consist of function definitions and function applications (and little to nothing else). Runtime of a program (which is a term) means evaluating the term. For an application term (which you can think of as $f(a)$), both $f$ and $a$ are also terms. Evaluating $f(a)$ may entail evaluating $f$ and/or $a$. The principle of leftmost reduction (lazy evaluation) means that $f$ is evaluated before $a$. Let us say $f$ is the function defined by $f(x)=x+x$ and $a$ is the term $1+2$. Then, lazy evaluation proceeds with the following steps:
(At least if $+$ does not have a definition of its own. If it has, then after 2. the definition of $+$ will be expanded.)
If, on the other hand, $f$ is defined by $f(x)=0$, the evaluation is
You notice that $a$ is not evaluated (because it is not needed), hence the connection to lazy evaluation. One final comment on this part: Lambda terms are in prefix notation, so "leftmost" actually means "outermost" when you think in terms of nested function applications.
In the second example above, lazy evaluation led to a speedup, because the term $a$ was not evaluated. In the first example, it led to a slowdown, because $a$ was evaluated twice. Actual lazily evaluated functional programming languages usually (again, I am not familiar with Clean) make the following optimization: Terms are not stored as syntax trees but as syntax dags. That is, in the first example, the two $a$ in term 2 would be the same node, also shared among all other positions of all other terms into which $a$ might have been copied. As soon as lazy evaluation dictates that $a$ is evaluated, the result is stored in place of the node $a$, becoming immediately available at all other occurences of $a$. The effect is a bit like memoization.