것으로 is basically just 로 being used with another grammatical device (specifically, turning a verb into a noun). In this case, 로 is more like "as".
급격히 rapidly. sharply
증가하다 to increase, to grow
증가한 것 an increase, growth
증가한 것으로 as something that has increased
나타났다 appeared, to have appeared, to have shown
The sentence translates to something like: As the number of smartphone users has gone up, the number of mobile shoppers has shown rapid growth.
I highly recommend studying the grammar relating to ~ㄴ 것, ~는 것, ~ㄹ 것 if you want to understand how the grammar interacts with the words.