When somebody tells you that ‘I look forward to speaking with you’ what they actually mean is that they are expecting to have a conversation with you in the near future. (I am using ‘they’ -plural to mean he/she).

The phrase ‘I look forward to speaking with you’ is correct and is mostly used in formal communication at the end of the communication. The sentence is written in a formal manner but can be improved further by rephrasing it this way; I am looking forward to speaking with you.

The (`-ing) in the looking and the (`-ing) in speaking sounds better than when the sentence reads ‘I look forward to speaking with you’.