。
Help me peel the eggs.
Literal
Egg [possessive-の] shell-peeling [object-を] help.
からむき ('shell-peeling') is a noun compound: から ('shell') + むき (masu-stem of むく, 'to peel'). Masu-stems routinely function as nouns in Japanese — a productive nominalization pattern. 手伝って is the te-form of 手伝う used as a soft casual request.