Whether you find this saying true (but pointless), merely a myth, or both, depends on your viewpoint about its origins.
In the Northern Hemisphere, March marks the shift from winter to spring, so by definition one would expect that conditions would begin cold and stormy and transition to mellower, warmer weather by month's end. If that doesn't strike you as especially helpful, you can look to the night sky for another explanation offered by the proverb's proponents: March kicks off with the constellation Leo (the Lion) on the eastern horizon at sunset but comes to a close with Aries (the Ram) on the western horizon. By their lights, it's a star thing.
Both reasons make the saying technically true but useless, which may explain why some versions add a few key words: "If March Comes in Like a Lion, It Will Go Out Like a Lamb." In keeping with the usual pattern of weather superstitions, this maxim makes a prediction: Rough weather during the first of the month means pleasant conditions at its close.
Whatever the proper version, we must sheepishly point out that there's no correlation between inclement weather at the beginning of March and pleasant weather later [source: Hambling]. But please don't bite off our heads about it.