I already know that the difference between v~いる & v~います is just that one is in the formal way and the other one in the informal. But I can't find nothing about v~いて.
I'd like to preface this response by saying that I lack the (academic) lexicon to explain this in linguistic terms, so I'm going to attempt to explain it the way that I've learned and most easily understood it. If anyone would like to critique/add better nuance to this response, I'd be more than happy to edit it to incorporate that information.
As it seems that you're somewhat familiar with ~ている you probably know that it's often used to convey an action in process, such as 食べている (is eating) or 歩いている (is walking). However, this is typically not the case for verbs that describe a change of state, such as やせる, 太{ふと}る, and (though slightly more difficult to understand) 行く.
In the case of such verbs, the nuance changes from "action in process" to "resultant state of continuing that action." e.g. If you continue to lose weight you will be thin. To generally understand what verbs fall into this category, here's a simple test: Is the answer to whether or not they've been done always going to be either "yes" or "no"? In the case of やせる (to become thin), you have either lost weight or you haven't. So if you use the present tense (その人はやせる) the nuance is that of "that person will lose weight"; the past tense, accordingly is "that person has lost weight." By using the -て form here we gain access to the middle ground of "that person is in the state of lost weight (read: skinny), but we're not commenting on whether or not this is a change that has taken place or just the normal state of affairs for them."
All that said, though I also learned the verb やせる early in my Japanese education, in retrospect I don't think it and 太る are great verbs for describing people's weight unless you're commenting on a change like "Oh, you've lost so much weight!" (...and even then it can come across as impolite depending on to whom you're speaking!) Nowadays I'd be much more likely to use something like 細{ほそ}い (slender) if I were trying to compliment someone.