。
She speaks to me, calling me Tom.
Literal
She [topic-は] me [to-に] Tom [quotative-と] say speaks-to.
話しかける ('to speak to, to address, to strike up a conversation with') is a compound of 話す ('speak') + ~掛ける ('initiate toward'). The ~掛ける suffix appears in many compounds for actions directed at someone: 呼びかける ('call out to'), 笑いかける ('smile at'), 問いかける ('pose a question to'). Here the form トムと言って ('saying "Tom"') frames the addressee being called by that name — apparently the speaker is being mistakenly or playfully called Tom.