The infinitive form of this verb is "eten". To make the third person singular in Dutch you drop the ending -en and add a -t. That is what happens with "drinken": -en +t = drinkt. The problem with "eten" is that eten: -en + t = *ett. But according to Dutch spelling, you won't repeat the "t" in the end, so "et". But since it is a long sound finishing with a consonant, you have to double the "e", hence "eet". Here is the thread about the present tense in Dutch: https://www.duolingo.com/comment/3762671 and the one about spelling: https://www.duolingo.com/comment/3733043
Another way of thinking about this is that the same thing happens in German and Old English.
Think of the -t verb suffix in both Dutch and German as English -th, thus English 'eateth' is regularised, but it was not always so; the -th merges with the root of the verb, so for Old English, instead of etþ (from etan, to eat), it would just be 'ett', however, this also causes umlaut, so the vowel changes it to 'itt'.
The same happens in German with 'isst' from 'essen', Dutch is more simple here in that it doesn't change, but the t in the stem merges with the -t for the suffix in the same way.