"1 year with possibility of extension" - what does this mean?

I'd probably read it this way:

I have funding for one year for sure, and hopefully for additional years, but it might depend on the success of future grant proposals, so I can't make any promises. But assuming that my funding is successful, and the postdoc is continuing to do a good job, I'd be happy to keep them for 2-3 years.

In some cases, postdocs might be funded by "hard money" coming from within the university; this is common in math, and such positions often have names like "Famous Person Assistant Professor". In this case, funding can be assumed to continue (barring a financial catastrophe for the university) and it means more like "2-3 years assuming you do a good job".

In my experience in the U.S., postdocs are exclusively for limited contracts, usually 1 year. Any departure from that or a longer initial period would be unusual, and would likely occur only in cases where the postdoc is bringing their own funding.

Therefore, you can't take much out of this information: in fact, the more meaningful part is probably "with the possibility of extension for another 1-2 years" - this indicates that if things go well for both you and the lab, there is a possibility of extension. However, it says nothing about whether the funding for those subsequent years is secure or not secure.

The alternative would be for a position to explicitly say that it is limited to 1 year with no extension.

I think @NateEldredge's answer is a suitable reading, but I disagree that the wording implies that funding is currently only secure for 1 year. It's simply that post doc positions are only offered for 1 year at a time.

I personally feel that one-year contracts are terrible both for postdocs and for the PI. Both have to invest a lot of time and effort in getting the project started: relocation, learning the rules of the new lab and where the coffee machine is, getting up to speed with the subject and methodology. Yet, the duration of one-year contract is not sufficient to get a published paper by the end of it, so no-one has an evidence to demonstrate success to support their next grant / job application.

Unfortunately, with the decline of research funding and increased competitiveness of academia, one-year contracts become a norm; even less-than-a-year contracts are not unusual, allowing universities to charge the same overheads but save the costs of annual leave. So, I would read the statement about an extension as:

I know one-year contracts are terrible; I wish I had funds for 2-3 years; I personally would not want to be in a position when I have to apply for such short-term job. To sweeten the pill I will mention a possible extension here --- this makes my ad to look a bit better and does not cost me anything.