If a private (one owner) LLC that was recently acquired by a private investment company offers "a grant of employee incentive equity in the company in the form of Class M units representing x.xx% of the total common equity of the company as of the day of the grant" what does that mean in general terms? Should the employee expect x.xx% of the company's sale amount (valuation) if it were to sell in the future (taking into account a vesting schedule of course)?
With LLCs, the operation agreement can define different shares for different kinds of income or equity, and different partners may be treated differently. In essence, you can end up with a different stock class for each partner/member. So you need to read the grant document and the OA really carefully to know what you're getting. You may want to have a lawyer read through it for you. This may be way more complicated than classes of shares in a corporation.
At the most basic level, the employee is getting a share of ownership in the company and would get a percentage of the sales price. That said, as littleadv alluded to, different share classes have different priorities and get paid in different orders. In a bankruptcy, for example, some classes almost never get paid in practice because they are so far down the ladder of priority.
The first step you should take would be to try to clarify what you are getting with the company itself. Failing that, contact a financial professional or an attorney in your area who can read the terms and give you a better understanding of the contract before you sign.