For some reason it feels strange to me to go right from すぎて to 私.
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Judging from your comment I think your main problem with understanding this sentence is in identifying the hidden pronoun/name. It sounds like you want it to say that the act of talking fast is difficult for you to understand, but this is not the case.
In fact 話し方がちょっと速すぎて is an abbreviation for something like 彼の話し方がちょっと速すぎて (insert name/pronoun of your choice in place of 彼). As usual Japanese omits reference to the subject when it is obvious from context. So the whole sentence is
His/her/Tanaka's way of talking is too fast, and it's incomprehensible to/for me.
So the て form just joins the two clauses with 'and' in the usual way.
I think the は acts as a contrast marker in this context. Most people can understand him but to me he is incomprehensible.
Edit: Another thought on what may be confusing you is that に is often used with すぎる. For example 私には高すぎる = It's too expensive for me. This に is not the に used in your sentence. That に is associated with 分かる and not with すぎる. If you wanted to use this construct you could say something like 話し方が私にはちょっと速すぎる, though I can't see a way to fit the verb 分かる into this construction.