I understand the contribution margin ratio = (Selling price - Variable Costs) / Selling Price.
Is it possible to use the contribution margin ratio and a profit figure to find a Fixed Costs value? I don't think I have a full understanding of what the CMR really is...

No - you need another piece of data, such as revenue.

Starting with Profit = Revenue - Costs, you can manipulate as follows:

Profit = revenue - (variable cost + fixed cost)
Profit = (Rev - Variable cost)*Rev/Rev - Fixed cost
Profit = CMR * Rev - Fixed cost
Fixed Cost = CMR*Rev - Profit

From this you can see that to determine fixed costs you need to know revenue in addition to CMR and Profit.

Here's an example - 2 cases that have the same CMR and profit, but different fixed costs:

Case A) Rev = 100, Variable cost = 30, and fixed cost = 50. Here CMR = (100-30)/100 = 70% and profit = 100- 30 - 50 = 20.
Case B) Rev = 50, Variable cost = 15, and fixed cost = 15. Here CMR = (50-15)/50 = 70%, and profit = 50 - 15 - 15 = 20.

So you have two cases with equal CMR and profit, but different fixed costs.

ABC Bread sells a box of bagels with a contribution margin of 62.5%. Its fixed costs are $150,000 per year. How much sales dollars does ABC Bread need to break-even per year if bagels are its only product? I know the answer is 240000. What I am trying to figure out is how they got the answer

Learn and memorize this equation:

Profit = Rvenue x Cont margin% - Fxed Cost

Rearrange:

Revenue = (Profit+ Fixed Cost)/margin%

At break-even profit = 0. You have values for fixed costs and margin %, so now solve for revenue.