Begin and start have meanings that overlap (and also overlap with commence, which is more formal).
In many sentences there is little difference:
"The sale (starts/begins/commences) on Monday".
There are some uses for which only start is used.
Start can be used as noun ("the start of the race), and start has an older meaning which is "to suddenly move" For example "John started when the car outside backfired" (= he was startled). Also When you set something in motion you start it. "Start the engine", or figuratively "Start a business".
To "start a conversation" is to (metaphorically) set it in motion. Also there is an idiomatic pattern "I couldn't begin to (do something)", which means "I couldn't (do something) at all. Put these together and you get "I couldn't begin to start a conversation" = "I couldn't start a conversation at all"