What's the meaning of fasts, is it only a physical deed of abstain from food and drinks? Because often in scriptures it's listed or placed in context of prayer, repentance, mourning or done to show a certain depending on HaShem. What has a physical deed of abstaining from food and drinks has to do with all of this? It helps us humble ourselves or afflict ourselves but where does this lead to, what is it's meaning to do this through fasting?
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R' Hirsch writes in Horeb that the purpose of all fast days is Teshuva, and gives the following explanation (ch. 33, Dayan Grunfeld's translation):

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> Fasting, i.e. abstaining from all kinds of nourishment for one day, should help in mastering the animal in man, in calling a halt to striving for self-gratification and in showing that a firm will can well remain the victor. It should also bring home the thought whether what made it possible for us to control our impulses and cravings - or, at any rate, the most pressing ones - for one day, could not make it possible for every day; and if we cannot control all such impulses, then at least we should shun the impure, the exuberant and the forbidden. Fasting, therefore, should contribute towards the eradication of one of the sources of sin, pleasure-seeking.
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Fasting is meant to demonstrate to ourselves the ability and purpose of not indulging in the physicality which leads to sin.
Fasting is a way of attaining a more spiritual state/frame of mind by disconnecting from physical pleasures. This more spiritual state is more conducive to Teshuva. See Nesivos Olam: Nesiv Hateshuva chapter 7 (beginning from the second paragraph) and Derech Hashem part 4 chapter 8:5 (regarding Yom Kippur).

However, today it is not recommended to fast (apart from the biblically/rabbinically mandated fasts) because it weakens the body and impairs the ability to think and learn Torah (Krina D'igrissa 1:16,17).