This is because she used a spear with dragonglass head.
Dragonglass is a common name in Westeros for the substance known as
obsidian, a form of volcanic glass. Along with Valyrian steel, it is
one of the two known substances capable of killing White Walkers.
Dragonglass is a substance that Children of the Forest use in their weapons.
According to legend, the Children of the Forest also made use of the
substance. Samwell Tarly discovered, by chance, that dragonglass is
one of the few materials that can harm or kill Others. According to
legend, the Children of the Forest would provide the Night's Watch
with weapons made of dragonglass specifically for that purpose.
Note that we've seen a White Walker killed by dragonglass before. As stated in the citation above, it was done by Sam in Season 3 episode Second Sons.
Later Sam and Gilly met Bran and his friends in Nightfort castle at the Wall. They told them about the event and Sam provided Meera with some dragonglass.
This means that she was aware of its properties and might have incorporated it into her weapons.
Alternatively, as in the scene we see Meera simply picking up the spear from the ground, it might have been one of the spears belonging to the Children. In the same scene they can be seen using spears, fighting wights and attempting to kill a White Walker.
Jon killed the White Walker using a sword made of Valyrian steel. For a detailed explanation see this question.