Short answer: Because 食べて is used to connect another clause to the sentence. This can’t be done with just 食べる.
Long answer:
The left (食べる) is either the plain form or attributive form of 食べる.
Plain form is the form of a verb which represents an event which has not yet completed and is usually used at the end of a sentence.
Attributive form is the form used when you are describing a noun. For example 食べる人 is a person who eats.
When 食べて is used, you should expect it to be connected to another clause. If you were to see 食べて人, 人 would not be being described by 食べる, but rather人 belongs to the next clause that is being connected after the て particle.
On the right (食べて) we have the conjunctive form of 食べる, which is 食べ, followed by the て particle. Even without the て particle conjunctions can still be made, but this is more of a literary thing as it’s a part of classical Japanese. Bunpro has a grammar point on this here. A very important thing to note is that when attaching the て particle to the conjunctive form a euphonic change happens. This is why the て form of 座る is 座って and not 座りて. In fact that’s the whole point of the grammar point that this thread is about.