SnagIt uses predefined trim edges to get its effects, and is a raster (pixel) program - you said you're pulling your word cloud into Illustrator as editable paths - so either it's coming in vector or you're Live Tracing it to become vector.
Once you're in Illustrator in an all-vector scenario, it's a bit more complicated, but you've far more control - you will group your disparate text and paths together, you will draw your cloud edge shape, and then you will select both the group underneath and the cloud shape above and right-click - choose "create clipping mask" and you'll be in business!
Grouped text underneath, cloud shape on top (coloured pink for clarity) - select both
Aaaand... Bob's yer Uncle!
Hope that helps!