Could anyone please explain how to determine the direction of the mass transfer [...]
In order for concentrations to change (given constant volumes and no chemical reactions), there has to be mass transfer. If the concentration in a solvent increases over time, mass transfer is to that solvent, if it decreases over time, it is away from that solvent.
So you have to look at the slopes in the diagram at the different time points. For the organic solvent, the concentration increases at both times, so mass transfer is from aqueous to organic phase.
This question is not about what you expect to happen, but what you observe to happen. It is great to check afterwards to see whether your observation matches what you expect, but you should interpret the observation in an un-biased manner. If you don't, there might be a temptation to see what you expect to see.
[...]and how it can be the same at t1 and t2
The slope has the same sign, so the direction is the same. This is quite commont - most processes don't oscillate as they approach equilibrium. In most cases, you see large changes at the beginning and smaller and smaller changes as you approach equilibrium. This is exactly what we see here.
As for why equilibrium is not when concentrations in the two phases are equal, see the answer by Night Writer.