There's a discussion of the difference between verb+出す (dasu) and verb+始める (hajimeru) over in the comments on -hajimeru. The consensus is that verb+出す is more of a sudden or explosive starting (like busrting into tears), while verb+始める is just simply starting.
Note also that this use of 出す is NOT the same as the standalone verb 出す, which means to take something from a more enclosed/private place or situation and put it in a more open/public one. 出す only means "start to do" when it's tacked onto another verb stem... and even then it may still have the above meaning:
取り出す - take (an object) out (of a container)
追い出す - chase (someone) out
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