To people wondering what 'sağ olun' means: -It basically means teşekkürler/thanks -But in other way it means 'be alive' -At the same time soldiers use 'Sağ Ol!' to reply to comrades(After saying Nasılsınız?/How are you?) It can also get used in different sentences in different meanings but I guess these are the primary uses for beginners.
So, to say it another way, it is always -grammatically- correct to leave them off, but pragmatically, you would use them in disambiguate? I know that there may be a really subtle (or maybe non-existent) difference between disambiguating and emphasizing, but I figured it was worth asking. It'll make my brain feel better at the very least.
The difference is due to the imperative (i.e. command) nature of "sen ye!". You are right that the word-for-word translation for "sen ye" is "you eat", but in English, if you want to command someone to eat, you probably wouldn't say "you eat!!!", you would turn to the person and say "eat!!!". (or, "iss!!!" in German)
In contrast, "sen yersin" is not imperative. It is a complete sentence that means "you are a person who eats". So "you eat" is the appropriate English translation. (or, "du isst" in German)
Same with this. "Sen içersin" means "you are a person who drinks". So the appropriate translation is "you drink". (or, "du trinkst" in German)
Ben içerim (I drink)
Sen içersin (you drink)
O içer (he/she/it drinks)
Biz içeriz (we drink)
Siz içersiniz (you drink)
Onlar içerler (they drink)
The verb tense (the -er part of the ending) follows what is called two-way vowel harmony and the personal endings (the -sin/-im) follow 4-way. I wouldn't stress out too much about this yet. Just memorize these verbs that you come across until you get to the present continuous skill :)