To start with you need some property which all Boolean functions built using only $\land$ and $\lor$ have in common. One that works is this: any such Boolean function evaluates to $1$ ("true") provided all the variables in it are given the value $1$.
Once the above property has been shown, any simple particular Boolean expression without the above property, such as $r=p \land \lnot q$ is seen not to be constructible from $\land,\lor$ since (for this example) $r$ comes out $0$ (false) when $p,q$ are each $1$.
Establishing the property mentioned above can be done either by common sense based on properties of $\land,\lor$ or else more rigorously by use of strong induction on the length of the Boolean expression, together with the recursive description of how longer Boolen "and/or" expressions are built up from shorter ones. For example a single variable is an "and/or" Boolean expression, and if $A,B$ are "and/or" Boolean expressions so are each of $A\land B$ and $A \lor B.$ How much detail to give in such a proof depends on the rigor level for whatever course the exercise is for.