What are you defining as a siege, if not encircling a fortified position to wait and force a surrender through attrition? That is the only use of "siege" I've come across in my studies.

It seems like you're confusing besieging a fortification with taking it by assault or by storm? I'm going to go ahead and assume that's the case.

So why take the risk of assaulting rather than starving the enemy out?

There are two main reasons.

First, think about the situation of the defenders compared to the attackers.

The defenders are sitting there in their castle. They've got stocks of food and water piled up. A castle might not exactly be cosy, but they have a solid roof above their heads.

Meanwhile the attackers are camped out around the castle. They're foraging for food, and as time goes on they'll have to forage further and further. They're living in tents, exposed to the elements. Their sanitation is rudimentary. Besieging armies regularly suffered horribly from diseases like dysentery running through their camps (eg. Henry V's army at Harfleur). Throughout history disease has often been as deadly an enemy as battle.

Second, think about time.

As a commander, do you have the time to starve a fortification out? Can you keep your soldiers for long enough? Your soldiers may start leaving, or you'll start running out of money. And what's the enemy doing in this time? Your forces are steadily weakening sitting around that castle, while your enemy has time and space to build up their forces and prepare for a counter-offensive.