I recently became a freelance and there’s something I’ve already noticed: people, most of the time, will choose the cheapest proposal.

They prefer to pay to someone that will take on a free wordpress template and change the text, costing them just a couple of hundreds, than go with a custom solution for their business.
Yes, a custom design and development is not cheap. But in business, just like with your electronics or anything else, if you buy cheap, you'll buy twice.

I've already heard many stories of business going south, and most of them failed because they chose the cheapest guy to do the job...
Situations where the clients paid for their projects and received a product full of bugs, where most of the things didn't even work, and the developer stopped replying to their emails or skype calls. Or worse, where the clients paid for the work and got nothing. Yes, nothing.
In other cases, they get the whole job done, but they don't see any improvement on their business, and they see themselves in a situation where they have to spend more money, in order to fix the real problem.

A lot of people want a website just for the sake of having one, and clearly they haven't figured it out yet. When they come telling me that they want a website for their business, it's not actually a website they want. What they want is to have more sales, stand out among their competitors, be the number one on the market, and grow their business. That's what they want! And every business is different, you can't use the same solution for every one! If that worked I would use the same template for every website I made!

If something isn't working for their business you have to figure it out why, and come up with a solution that works. And that won't come cheap. They have to understand that you're not being paid just for the website, you're being paid to deliver a complete solution for their business.

That's why I see a red flag when a client comes up with exactly what they want, and leave no room for changes or improvements. When they want someone "do the work", I get suspicious, which may lead me to refuse that offer.
In those cases I talk with the client and let them know that if they want someone with experience to make their idea a reality, who will help them with the design, business strategy, marketing and development, who will be available, and advise them along the way... then I am the person they're looking for. Otherwise, they'll have to find someone else.

Or like Paul Jarvis say:

Sometimes you need to gently remind your client that they’re paying for your expertise, not hiring a monkey that says “yes sir/ma’am” to every request (if monkeys could talk, that is).

That is something that I do, and probably will be doing for a lot of time. And it's not the clients fault. Sometimes they just don't know any better.

For me the most important part of getting a proposal approved, is in the communication with the client. Listen to their needs, how can you help them and why you are the right person for the job. It's also important for them to know that if they need to reach you, you'll be available. Although, it's a good idea to set up a schedule for replying the emails, otherwise you'll be constantly interrupted.

Keep things clear from the start; set the clients expectations to avoid "this is not what I asked for", or "I was expecting a different thing" kind of reactions. Set up a proposal where you define the problem, how you'll fix it, and why that will work. If you were in a similar situation it might be a good idea to show the results and prove that your solution works, or that you already have some expertise in the field.
It's also important you state your terms and conditions for your own protection. Set up the schedule for the work and the payments, describe your process, your rights, kill fees, and always date and sign the proposals.

The number of websites for freelancers have increased a lot in the last years, and most of them work exactly the same way: you set an hourly rate, or your value for the job, and you'll be competing with other freelancers for that value. They don't usually reward for your problem-solving skills. If you set the lowest price, you'll probably be assigned for it.
That is sad, but it's also our job as freelancers to show the clients that we are the right person for the job, and show them that the price they're paying will worth every cent.