Where does “beszélte” come from?

beszélte (Hungarian) comes from Hungarian beszél, from Hungarian beszéd, from Proto-Slavic besěda, from Proto-Slavic bez, from Proto-Indo-European *bʰeǵʰs.

Ancestry of “beszélte”, step by step

Step	Language	Word	Meaning
1	Hungarian	beszél	to speak, talk; to matter, to count, to have...
2	Hungarian	beszéd	speech
3	Proto-Slavic	besěda	sitting outdoors; an outdoor gathering;...
4	Proto-Slavic	bez	without
5	Proto-Indo-European	*bʰeǵʰs	—

Every word from Proto-Indo-European *bʰeǵʰs →Every word from Proto-Slavic bez →Every word from Proto-Slavic besěda →