As far as I'm concerned, yes, it is true.
I've tested this multiple times and I've come to the conclusion that turning Wi-Fi off is counterproductive. Battery life has always been better for me with Wi-Fi turned on (unless cellular data is also turned off, in which case Wi-Fi being off/on is irrelevant).
IMHO the idea that turning Wi-Fi off has any noticeable power saving affect is an urban legend that won't die, because few can imagine that turning something on could use less power. Logic suggests anything that is turned "on" should require more power. In this case that's just not necessarily true.
Consider your device's ambient light sensor. It actually creates power (like a solar cell) when it is exposed to light. Otherwise it just sits there and does nothing, which uses almost no power at all. Wi-Fi is similar. It sits there, waiting for electromagnetic waves at a certain frequency to induce electricity in the Wi-Fi antenna. This is called passive Wi-Fi AP discovery and it requires almost no power at all.
The Wi-Fi standard also supports an active Wi-Fi AP discovery method. When implemented correctly this should never be used however, except when you're in the Wi-Fi settings page and telling the device to scan/refresh its list of available W-Fi networks.
On the flip side, if you are in the vicinity of a Wi-Fi AP that your device could connect to but doesn't (because you have Wi-Fi turned off), then you are forcing WP to do it's bi-hourly live tile updates and all notification sending/receiving over the cellular network! Transmitting to a cellular tower is one of the most power intensive things a smartphone could possibly do, in which case leaving Wi-Fi on would have saved you a lot of battery power.
That is what the message is telling you. It's not really correct that leaving Wi-Fi on saves you battery power. It's that a smartphone that is forced to use the cellular network when it could have used Wi-Fi uses more power than is necessary.
Originally, WP didn't allow you to turn Wi-Fi off at all. That feature exists now, not because it is technically necessary or even a good idea, but because people have just come to expect it and feel the phone is otherwise missing a very basic feature.