How is SIESTA and TranSIESTA different from plane wave DFT codes?

Siesta relies on the LCAO method which is different from the plane wave (PW) formalism encountered in the VASP and QE codes.

A noteworthy difference between the two types of methods is the convergence of precision. In PW there is basically a single value (the plane-wave cutoff) that you simply increase to improve precision. In LCAO the basis set is more important since there are length cutoff's of the orbital ranges. One can always improve precision by increasing the orbital ranges but there are still many more parameters to fine-tune. It is vital to really look into the details of improving precision for individual LCAO codes.

TranSiesta is an extension for Siesta that implements the non-equilibrium Green function theory. The Green function theory relies on self-energies which are basically the equivalent of a semi-infinite (but not limited to) bulk parts. This is drastically different from PW codes and other codes with full periodicity. The semi-infinite replaces a part of the Hamiltonian with the exact bulk equivalent such that one is really simulating a semi-infinite bulk electrode connected to a device. For instance if we label a Gold ABC stacking with A, and a molecule with M, and create a simulation cell comprising A-M-A.
In PW this would equate to simulating:

 ...[A-M-A]A-M-A[A-M-A]...

with each [ ] repeated infinitely.

In NEGF theory one would replace A with the equivalent bulk part thus simulating something like:

 ...[A][A]A-M-A[A][A]...

where each [ ] is repeated infinitely.

You'll notice the drastic change in the full system. This puts certain constraints when performing the simulation since it is required that the simulation cell's A parts are converged towards the bulk properties (i.e. same potential). Otherwise one would create an artificial interface between the simulation cell and the exact bulk properties of A.

As for transport in either method there are implementation details that make the LCAO method much easier to implement. The fact that the basis set is local means that one can efficiently calculate the surface self-energies[1]; these are the basis of "bulk" transport. The reason is that one can easily partition the system into "electrode" and "device" regions. And thus efficiently attach bulk surface self-energies, see above.

In PW codes this partitioning is not as simple due to the non-local basis set, see this question While in fact one can implement transport calculations in PW [2] it seems it is more difficult and not as standardized in codes.

However, before endeavouring into transport calculations it is highly recommended to really understand the basis of Siesta (LCAO) and fine-tune the calculations, then secondly understanding the requirements of the interface potential.

References

1. M P Lopez Sancho et al 1985 J. Phys. F: Met. Phys. 15 851
2. Garcia-Lekue, Aran, and Lin-Wang Wang. "Elastic quantum transport calculations for molecular nanodevices using plane waves." Physical Review B 74.24 (2006): 245404.