LLC is a legal designation where the main benefit is liability protection.
For taxation, an LLC can be either a sole proprietorship (file schedule C within 1040 individual return), a partnership (the default with 2 or more owners) or an S-corp.
The S-corp is really only a benefit if you make a decent amount above what would be considered a "reasonable salary". Reasonable salary is an incredibly subjective thing, but it is what the definition is in the IRS code. Depending on the scope of the work, the expertise level required, the time commitment and revenues brought in, the reasonable salary can vary greatly.
The reason you have to beat the "reasonable salary" amount is because of how the payroll taxes are handled. As a sch C filer, you have to pay self employment tax on the entirety of the business income. That is an automatic 15.2% tax that isn't able to be reduced by any itemized personal deductions, exemptions and even some of the credits. That's because it is the replacement for what the employee and employer pay together for FICA taxes (SS & Medicare).
As a sole owner of an S-corp, you are required to pay a reasonable salary to yourself and that salary is treated just like an employee's salary with all of the taxes taken out and then the business has to pay in its portion of the payroll taxes as well. Now the salary/S-corp has one component that the sole proprietor does not and that is unemployment taxes, both federal and state...which depending on your state may not be insignificant.
So why would you want to go with an S-corp if you have to pay the extra for unemployment? Well because of what happens to the income in excess of your salary amount. That excess income is no longer subject to self employment tax, so therefore, you save 15.2% on everything over and above your salary. If you only make an extra 5k, it may not be worth it be paying the unemployment plus having to pay to have an extra tax return prepared. But if you make $20K above the "reasonable salary" amount, that means savings of about $3K (less the unemployment and tax prep costs).
As you continue to make more over and above the salary amount, that tax savings really adds up because the unemployment and tax prep costs aren't increasing in proportion. Once you get up to six figures of net business income on a schedule, you are stupid to not be running it as an S-corp and throwing away thousands of dollars.
The true cutoff point is really dependent on what that "reasonable salary" amount is for the industry and job in question. I've worked on company's where $24K was a legit reasonable salary and I've seen cases where it should be around $75K. It really is a wide range.
ETA: forgot to mention the extra expense of having to set up payroll tax accounts with the IRS and state and having to keep track of and remit those taxes with reports on a monthly/quarterly/annual basis. It is a bit of a headache and expense...there are cheap payroll services out there like intuit payroll, but you still have to log in and do it all yourself.