Translation:Grandmother lives up on the high mountain.
"Höga" is the definite form. Definite forms are used when the adjective is right before a definite noun.
In most cases definite forms are identical to plural forms.
Note that it's only used when the adjective is right before the definite noun, and not just whenever the noun it describes in the definite.
"Det höga berget"
"Berget är högt"