In a nutshell what it can mean is that your employer will terminate your current employment contract and you will enter into a completely new employment contract. You will be a "local employee" which means, as you've said, that all your employment terms, rights and obligations will be governed under Swiss law. Since you are entering a new "local" contract you may find your salary, benefits and level change to something more in line with the Swiss market. You need to check this to make sure you won't be disadvantaged versus what you get now.
Not least to consider is that you will be paid in Swiss francs under a local contract and therefore subject to FX risk if you have obligations or outgoing in dollars. Sometimes this may benefit you. Sometimes you may lose out.
You would also normally cease to be covered under the US company pension scheme and social security scheme, and instead you would join the Swiss pension scheme. If any of your current benefits (like pensions, promotion and pay) are negatively affected by length of service this is something to be wary of and to look into very carefully.
It's not always the case, but it's usual, that international transfers on a local package don't qualify for expat relocation benefits like housing assistance, cost of accomodation searches, help with utilities, health insurance, home leave (an annual round trip airfare home) etc. Even accountants fees are sometimes paid because, if you are a US citizen, you will continue to pay US taxes and will therefore have two tax returns to complete and deal with double tax claims.
Expat packages are really designed to keep you under your home employment contract and insulate you from the consequences of your move through tax equalisation and cost of living adjustments to your compensation so you aren't disadvantaged financially through the overseas move. Bear in mind your cost of living will be higher in Switzerland so if you're on a local package you need to understand how the local package will affect you.
If and when your new employment contract comes to an end and you want to return to the USA, then your company will have no obligation to repatriate you, which means they will have no obligation to offer you a position back in the USA (perhaps that would be "difficult" because they will have replaced you) and you may have to pay all repatriation costs, like lease break penalties, removals and airfares unless you have negotiated otherwise before you leave.
There are lots of variations on the above themes. Some employers are more generous or open to negotiation with others. I recommend you discuss this very carefully with your employer and get everything in writing before you commit.