What is the "maximum power" of an antenna?

It actually stands for the maximum power that you can push through this antenna.

This typically has three reasons:

• When designing the antenna, there's some point at which the electrical field is maximal. You have to be sure that the electrical field still doesn't lead to dielectric breakdown (or excessive nonlinearities)
• Safety: Power -> voltage -> safe to operate in the specified environment (electrocution risk); also: power -> power density -> safe to operate (hurting people in the main beam); this is usually more of a system than an antenna aspect.
• Heat: No real-world antenna is 100% efficient. The remainder of energy is converted to heat, which increases antenna temperature, which might change or damage the antenna.

So, to evaluate these, you'd need to have a pretty detailed simulation of the antenna; typically, you'd want to actually test maximum power, not only simulate it, which will inevitably risk damaging the antenna you have.

When connected to a transmitter, this (or any) antenna is designed to handle transmitter power levels up to this specified limit without degrading performance or posing a safety hazard. There is no easy way for you to test this manufacturer's rating without destroying the antenna or risking harm.

It's just the maximum transmit power the antenna is designed to handle. If the transmitter ends up pumping more power than the stated maximum into the antenna, the losses in the antenna will overcome its heat dissipation capability and things will start to overheat/melt.

Note that it's specified at a particular ambient temperature, because running the antenna in a hotter environment will compromise its cooling, so it won't be able to take as much power.

Why do you suspect the datasheet is not right? Do you think it's underestimating the capabilities of the antenna? Or that it's overestimating it and the antenna will melt way sooner than at the rated power? Either way, the only way to check experimentally is to take a transmitter, make it feed the rated power into the antenna, and see whether things melt. Note that you will probably destroy an (expensive!) transmitter this way if the antenna fails.