In this case, the て-form of 歩く has a quasi-adverbal function, describing the manner of the verb 来る (来ます). The phrase 歩いて来ます means "to come by walking", as opposed to eg. バスで来ます "to come by bus" or 走って来ます "to come running".
More generally, the て-form is used to connect verbs together, often to indicate a chronological sequence (eg. 勉強して帰る "to study and then go home") or to form a semi-compound meaning. In the sentence you listed - 私は歩いて来ます - it wouldn't be possible for the basic form 歩く or 歩きます to occur in this position, because two verbs can't be simply placed together in this way. 私は歩きます来ます would be like saying "I walk come" in English - the two verbs don't make sense as a sequence.
If you want to list the two verbs together like that, there needs to be some grammatical feature connecting them, and the て-form is one of the constructions that can do this.
In both of the other sentences you listed, 歩きます occurs at the end of the sentence as the main verb, so there is of course no need to use the て-form.