What is a projection method? Quoting from Solenthaler et. al. Predictive-Corrective Incompressible SPH
 (ACM Transactions on Graphics, Vol. 28, No. 3, Article 40, Publication date: August 2009) (PDF link here)

These incompressible SPH (ISPH) methods first integrate the velocity field in time without enforcing incompressibility. Then, either the intermediate velocity field, the resulting variation in particle density, or both are projected onto a divergence-free space to satisfy incompressibility through a pressure Poisson equation.

What does the author mean by "project?? Is there a simpler way of understanding this operation? I tried reading other articles, but they go even more deep saying solenoidal vectors etc. I am looking for a simple explanation at first to understand the concept.
 A: This would be a better question for scicomp.stackexchange.com. But, in essence, the projection method refers to a two step process of integrating the velocity field. Incompressible solvers are designed to advance the velocity field so that it most acurately solves the discretized momentum equation. There are several methods available to do this, such as, the pressure correction method, the theta method and the pressure projection method. In the latter case, there are essentially two steps. 
Step 1:
In the first step, we obtain an intermediate velocity in the absence of a pressure field (that way we can solve it easily).
$\frac{\mathbf{u^*}-\mathbf{u^n}}{\Delta t} =$ all remaining terms in momentum equation - pressure terms
This gives you an intermediate velocity $u^*$ which because of the missing pressure term does not satisfy continuity (hence why he said " ... without enforcing incompressibility").
Step 2:
The second step brings the pressure term back in order to correct the divergent velocity field, $u^*$. This is the part where he says "are projected onto a divergence-free space to satisfy incompressibility through a pressure Poisson equation.", in other words:
The divergent velocity field is corrected with the corrector:
$\frac{\mathbf{u}^{n+1}-\mathbf{u}^*}{\Delta t} = -\mathbf{G}p^{n+1/2}$,
where $\mathbf{G}$ is the discrete gradient operator and the pressure field is computed with:
$Lp^{n+1/2} = \frac{1}{\Delta t} \mathbf{D} \cdot \mathbf{u^*}$ (poisson equation),
where $\mathbf{D}$ is the discrete divergence operator and $L$ is the discrete Laplacian operator. The superscripts indicate the time step, note that n+1/2 is at half time step.
So, first you calculate a divergent velocity field $\mathbf{u^*}$ which lives in a space that does not satisfy continuity. Next you correct that velocity by "projecting" the divergent velocity field onto a subspace which is divergence free (that is, the continuity equation is satisfied) using the pressure solution from the Poisson equation. They mean projection in the mathematical sense that you are making a transformation from one space to another.
