With SparseArray`KrylovLinearSolve[A,b, "Preconditioner" -> f] you can use an arbitrary function f as preconditioner. Here an example:
precdata = SparseArray`SparseMatrixILU[A, "Method" -> "ILU0"]
f = x \[Function] SparseArray`SparseMatrixApplyILU[precdata, x]
Other accepted values for the option "Method" are "ILUT" and "ILUTP".
More or less,
SparseArray`KrylovLinearSolve[A,b, "Method" -> "BiCGSTAB", "Preconditioner" -> f]
should then be equivalent to
LinearSolve[A,b, Method -> {"Krylov", "Method" -> "BiCGSTAB", "Preconditioner" -> "ILU0"}]
Other supported Krylov methods are "ConjugateGradient" (only for symmetric positive-definite matrices) and "GMRES"
See the documentation of LinearSolve, section Options, subsection Methods subsubsection "Krylov".
Using SparseArray`KrylovLinearSolve allows you to build your own preconditioners.
See standard literature on numerical linear equation for more information about what a preconditioner is. Basically, a preconditioner is meant to speed up the convergence of iterative linear solvers. Expanding on that would be out of scope for this site.