I am trying to understand the connection between cost structure of a business with idea of using contribution margin to make a profit.

The Total Contribution Margin (TCM) is Total Revenue (TR, or Sales) minus Total Variable Cost (TVC):

TCM = TR − TVC

The Unit Contribution Margin (C) is Unit Revenue (Price, P) minus Unit Variable Cost (V):

CM = P − V

The Contribution Margin Ratio is the percentage of contribution over sales, which can be calculated from the unit contribution over unit price or total contribution over total sales.

C/P = (P-V)/P = Unit Contribution Margin / Price = Total Contribution Margin / Sales

Contribution margin can be thought of as the fraction of sales that contributes to offsetting fixed costs. Alternatively, unit contribution margin is the amount each unit sale adds to profit: it's the slope of the Profit line.

PL = TR - TC
= (C + V)X - (TFC + V)X
= CX - TFC

where, TC = TFC + TVC if total cost = total fixed cost + total variable cost and X is the number of units.