Learn Farsi: من خویشتن را در این بیت پیدا کردم.

Khishtan is a formal, literary version of khod (oneself). While in everyday spoken Farsi you would say khodam (myself), using khishtan elevates the sentence. It sounds much more poetic and philosophical, which is perfectly fitting when talking about reading a verse of poetry.

Does khishtan need a personal ending like khod does?

No, and that is a key difference. While khod almost always takes possessive suffixes in everyday speech (like -am for myself or -ash for himself), khishtan usually stands alone. It automatically reflects the subject of the sentence, which here is man (I).

Why is the object marker ra attached directly to khishtan?

Because khishtan is the definite direct receiver of the action (the specific person being found). Since it does not take personal suffixes like khodam does, the object marker ra simply follows the base word khishtan to show it is the direct object of the verb peyda kardam.

What kind of verse does the word beyt refer to?

A beyt specifically means a couplet or a two-line verse in traditional Persian poetry. Because classical poetry is deeply woven into Iranian culture, talking about finding your own soul, life, or situation in a beyt is a very natural literary expression.