Step 1: Analyze Assertion (A).
The dissolution of many salts is an endothermic process.
For an endothermic dissolution,
\[
\text{Solute} + \text{Heat}
\rightleftharpoons
\text{Solution}
\]
According to Le Chatelier's principle, when temperature is increased, the equilibrium shifts in the forward direction to absorb the added heat.
As a result, more solute dissolves and the solubility increases.
Therefore, Assertion (A) is correct.
Step 2: Analyze Reason (R).
In a saturated solution, a dynamic equilibrium exists between the dissolved solute and the undissolved solute.
\[
\text{Undissolved Solute}
\rightleftharpoons
\text{Dissolved Solute}
\]
At equilibrium, the rate of dissolution equals the rate of crystallization (or precipitation).
Therefore, Reason (R) is also correct.
Step 3: Determine whether R explains A.
Since a saturated solution is in dynamic equilibrium, changing the temperature disturbs this equilibrium.
For an endothermic dissolution process, increasing temperature shifts the equilibrium towards the dissolved state, resulting in greater solubility.
Thus, the concept mentioned in Reason (R) provides the basis for explaining why solubility increases with temperature in Assertion (A).
Step 4: Apply Le Chatelier's principle.
Because dissolution is endothermic,
\[
\Delta H \gt 0
\]
and heat behaves as a reactant.
An increase in temperature shifts the dissolution equilibrium in the forward direction, increasing the amount of dissolved solute.
Hence, the dynamic equilibrium referred to in Reason (R) correctly explains Assertion (A).
Step 5: Final conclusion.
Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct explanation of Assertion (A).
Therefore,
\[
\boxed{\text{Both (A) and (R) are correct and R is the correct explanation of A}}
\]
Hence, option (1) is correct.