What country are you in? Retired@63 is correct, but that's not a common set-up in the US.
Full disclosure: I work for a US insurance carrier, one that does most of its business through independent agents and brokers.
In the US, what an agent does depends a lot on whether they are "independent" or not. Independent agents have contracts with multiple insurance carriers and find customers insurance through those. They get paid on commission; 10-15% of the premium, and possibly additional bonuses depending on volume. So, on one hand, the more you pay the more they get paid, but at the same time, moving your insurance is relatively low-hassle (for a financial service), and getting new customers costs a lot more than keeping existing ones, so they also want to keep you happy.
Non-independent agents are those that only represent one carriers: State Farm, Allstate, American Family, Farmers, etc. Those agents are essentially the sales arm of those insurance carriers. They also get paid on commission, but there's no picking among carriers. They just help you pick what's best for your situation from the coverage offered by that carrier.
I've never had to file an auto or homeowners claim, so I don't really understand that process. Do people usually go through their "agents" or directly call the insurance company hotline?
Usually, you'll want to report the claim directly to the insurance company first, and then to your agent. If you wanted advice on whether or not you should report a claim (like if you don't know if it's covered and don't want it on your claim history if it's not, or if it's just over your deductible and want to know what the impact will be), you could call an independent agent first with that question. A non-independent agent might be able to help with those questions, but they're essentially the same company, so you don't get the same independent advice.
After the claim is reported, your agent (either type) can help you navigate the claims process. For example, I had a theft claim involving laptops, and the claims adjuster very clearly didn't understand a specific clause around electronic equipment. I had no luck with him or his manager, but my agent called someone and got it straightened out.
How much does an agent ACTUALLY do?
A lot of what they do is answering questions and providing advice on what coverage you should buy. For independent agents, that should also include shopping your coverage around the carriers they are contracted with to see if they can get you a better deal, and presenting you with their findings. Plus, independent agents carry Errors & Omissions insurance, which can help you if they screw up.
As a hypothetical, let's say that I had told my agent that I had a lot of computer equipment I needed coverage for, and she recommended Carrier A. There's a theft claim and my equipment gets stolen, but Carrier A says it's not covered because of their electronic equipment restriction and I should've bought different coverage from them, or gone with a different carrier. The fact that the agent screwed up my coverage means that I can make a claim under her E&O coverage. (If, however, she had recommended Carrier B for my situation, but provided an option that Carrier A was cheaper but less coverage for computer equipment, then the error was mine.)
Those of you with good agents, how do you subjectively put a quantifiable monetary value on them vs. shopping around for insurance like many people do?
Insurance is a promise on a piece of paper. If you never expect to need the promise, then it really doesn't matter what's behind the paper, and even if you do, it's going to be almost impossible to quantify the quality of promise, you know?
So, personally, I don't put a whole lot of monetary value on my agent, even though I firmly believe in the independent agency system. If my insurance costs me an extra $200 a year, I'm betting that will save me more than $2,000 every 10 years. That'd be hard to do since I've only had a total of $2,000 in insurance claims in 10 years, so it's not a bet I'm willing to make.
I shop around for insurance every year, going direct to places like Geico and Progressive, and asking my agent to check for better deals. I used to go to the dependent agents too, but never got anything useful and it generated a lot of junk mail. Despite that, I have never found a better deal than the company I'm with right now through any channel. As a mustachian, I don't drive that often, and the company I'm with has an option for "pleasure" vehicles that are driven less than 7,500 miles per year and not used for commuting. Recently, other carriers are offering auto telematic options that might save me money by tracking my driving, but I would have to switch to them, pay more initially, and hope that eventually I see some savings, which doesn't make sense for me.
I pay $800 per year for my auto insurance (1 small car, two clean drivers, $1,000 deductible, $300k limit, full uninsured motorists coverage and comp/coll), and $1,600 for my homeowners insurance ($250k dwelling, $200k contents, $300k liability, $1,000 ded). (I don't have anything close to $200k contents, but it's a standard percentage of the dwelling costs.) Obviously, different parts of the country have different risks, but maybe this would help you.