Number of comparisons in Binary search
The recurrence is supposed to express the worst-case runtime of binary search on an input of size $n$ (the letter $T$ might stand for time). The range you have to search search is halved in each step, so you get the $T(n/2)$ term on the right side. 

Most of the time people only count the comparisons of the data, not the extra comparison for the loop condition, so it takes 1 comparison to halve the search space. For your analysis you might want to write a $+2$ instead of the $+1$, since you do count the loop condition. 

As we want to express the worst case runtime, we assume that the element is not in the array, so the recursion terminates only when there is just one element left. This provides us with the base case of $T(1)=1$.