。
Fill it up to the brim so no air gets in.
Literal
Bottle [possessive-の] mouth [up-to-まで] fully put-in [then-て] air [subject-が] does-not-enter [make-sure-ようにしなさい].
A multi-layered instruction. いっぱいに is the adverbial form of いっぱい ('full'). The ~ないようにしなさい combines negative purpose (ように 'so that') with the polite imperative (しなさい). まで marks the extent — up to the mouth of the bottle.