Payoff from an option contract

In period 1 the consumer of type $$\theta$$ selects an option contract consisting of an up-front fee, $$B>0$$, and exercise price, $$\bar{R}$$. The consumer pays $$B$$ at the end of the first period. In period 2, he realises his valuation, $$\theta$$, distributed on $$[0,1]$$ by CDF $$G(\theta)$$ with density $$g>0$$. His expected payoff, from choosing contract $$(B, \bar{R})$$, should thus be $$-B+\int_{\bar{R}}^{1}(\theta-\bar{R})g(\theta)d(\theta).$$ However, in the paper I am told that it is instead: $$-B+\int_{\bar{R}}^{1}(1-G(\theta))d\theta.$$ Are these two expressions equivalent? What does it mean to integrate a CDF in this fashion? Thank you.

When integrals look different than what pops into your head, often the reason is integration by parts. For your example note that $$\int_R^1 (\theta -R) g(\theta) d \theta + \int_R^1 G(\theta) d \theta = (1-R) - 0,$$ where the right-hand side is equivalent to $$\int^1_R 1 d\theta$$. Hence, the two expressions you consider are equivalent.
It's of the form $$\int u(x) v'(x) dx + \int u'(x) v(x) dx = \int u(x) v(x) dx.$$