What is the point of 'Soldier of the Quarter' boards?

It's a competition and you start to see the real prestige that comes from winning boards when you hit the Division levels and above. That's when the competition becomes fierce, the Soldiers you're competing against are the best their brigades have to offer and they are actually practicing. Winning boards comes with rewards. The Division board is held at the one or two star general level and being personally recognized by the general of your Division, Corps, or MACOM comes with some great promotion potential. When that General meets you, the Best Soldier in the Entire Division, and asks you what your future plans are, you have the power of a division staff and the recommendation of a General to assist you. When they ask you what your future goals are and you say, Airborne, Ranger, OCS, ROTC, WOCS, Aviator, PCS etc, that General will look at their staff and say, "Make that happen". Next thing you know, you have your Brigade Commander asking if you need help putting your Green to Gold packet together, or whatever it is you want to do.
Plus, you'll learn a whole lot about Army knowledge in the process.

It is a way to set yourself apart/above your peers. It is a test of your skills and abilities. It is bragging rights, not just for you, but for your unit.
Why did you climb the mountain? Because it was there.

They help for overall career enhancement. You will have a record that you excelled against your peers. It quantifies that you went above and beyond just meeting the baseline standards. I don't know how the Army does it. The Air Force, typically at your level will go to an Airman who performs their job at a proficiency level well above their peers. They are the ones who can do their job without being instructed. They are the ones who are also doing things for self improvement like getting their associate degree though the CCAF and are typically going further to get their Bachelors. They are doing volunteer stuff like participating on the base honor guard for example. They are the ones who aggressively working toward their journeyman 5 level in their AFSC. I would say like in a squadron of 30 Airman. 20 will be average to above average. A small handful will struggle they might be able to do their job well, but don't take initiative. Of course there is the rare dirt bag. Then there will be like three or four who are well above average to their peers. They are the ones who show that they are capable to move further in their career. Those are the ones who get those awards. In the USAF if an E-4 wins the wing level, they are typically step promoted to E-5. Those are the ones who if they put in a packet for a commissioning like a full ROTC scholarship or in some cases an appointment to the academy, they get it. Others who do not desire to go commissioned are typically getting groomed for upper level NCO leadership, like eventually to possibly be a Chief.