This is untrue as a general rule, though it can be closer or farther from the truth depending on market conditions.

To see why, imagine that every month you buy a supply of fizzlesprots from Acme Corp. Today is the first of February, so you eagerly rush off to buy your monthly fix. But wait! The government has just imposed a tax on all fizzlesprot purchases. Curses! Now you'll have to pay even more, because Acme Corp will just pass the whole tax on to you.

Now change "fizzlesprot" to "labor" and "Acme Corp" to "employee". Huh? You're an employer, not an employee? My world is turned upside down! Could it be that the narrative where You bear the full brunt of every tax and They end up paying nothing is wrong?

In fact, whenever an economic transaction is taxed, the buyers and the sellers split the tax based on who is more eager to buy or sell. Labor is no different. It's possible that, empirically, the employee usually pays more of a labor tax than the employer, but this is by no means guaranteed and I would personally expect the proportion to vary significantly between labor market segments.

(Wikipedia's article on tax incidence claims that employees pay almost all of payroll taxes, but cites a single paper that claims a 70% labor / 30% owner split for corporate income tax burden in the US, and I have no idea how or whether that translates to payroll tax burden or whether the paper's conclusions are generally accepted.)

For more details, consult your nearest introductory economics textbook.