、。
Smiling warmly, mother held out her hand to the baby.
Literal
Mother [topic-は], warmly [adverbial-と] smiling while baby [to-に] hand [obj-を] held-out.
にっこり is a 擬態語 (state mimetic) for a warm, pleasant smile — distinct from にこにこ (continuous smiling), にやにや (smirking, often creepy), and くすくす (giggling). The optional と that follows mimetics marks them as adverbs modifying the verb. ~ながら attaches to 微笑む ('smile') to express 'while smiling.' 差し出す is a compound verb (差す + 出す) meaning 'extend/hold out,' especially hands, arms, or objects offered to someone.