What is the meaning of "labor input" in the context of incentive theories?
Since the authors state that the total labor input is:
$$\int t_i(k)dk$$
the meaning of the total labor input in this case would be that it is the sum of all attention $t_i$ allocated over those tasks $k$.
For example if we would assume that $t_i (k) = k$ then the labor supply across continuum given by $[0,1]$ would be equal to $\frac{1}{2}$ because $\int k dk = \frac{1}{2} k^2 + c$ and when you evaluate it between bounds $[0,1]$ the area would sum to $\frac{1}{2}$ (note that geometric interpretation of an integral is an area under the curve that you are integrating).
You can graph it by making assumption about the function, for example following my assumption of $t_i(k)=k$ the labor supply can be plotted (using tikz in LaTex) as:

Where the total labor supply is the area under the curve bounded by $[0,1]$. Of course you might want to impose some less simplistic assumption on the function $t_i (k)$.