The boundary conditions are imposed by the 'boundaries' (i.e. the spatial position of the plates defining the extent) of the transmission line.
For example: visualise two parallel plates of (we'll assume perfectly conductive) metal extending to infinity in the XY plane, spaced (say 1mm) apart such that one plane was at Z = 0 [mm], and the other at Z = 1 [mm]. A TEM wave propagating between the plates is completely invisible to an observer sitting above them at (x=0, y=0, z = 5), for instance. Similarly (by reciprocity), the observer cannot influence the TEM wave travelling between the plates beneath his feet from his position either. (i.e. no transmission/reception/coupling to free space from the transmission line).
Why is this the case? Because the plates are perfectly conductive, there cannot exist any electric field component tangential (i.e. in the same plane) as the plates in the same place they are. If you tried to establish a field gradient parallel to the plate, it would be "shorted out" by the perfect conductor. (Perfect conductor = zero resistance = zero voltage differential irrespective of the induced current). Consequently, only an electric field component *normal* to the plates can exist in their proximity - which gives rise to the only propagating modes between them as the solutions to the wave equation describing the classic TE, TEM and TEM modes. A wave propagating in the free space (where we refer to it being unbounded - i.e. having infinite extent in the case of plane waves, or only loosely constrained in the case of optical Gaussian modes) outside the plates cannot pass through the plates for the same reasons. If the incident wave induces a current in the plate surface as a result of having an electric field component parallel/tangential to the plate, this current causes re-radiation (appearing as a reflection) from the incident surface. Nothing appears on the other side of the plate - it has imposed a "boundary".
I hope that makes some semblance of sense! Unfortunately, most introductions to EM propagation leap straight into a world of div, grad & curl without offering any intuitive insight into the processes going on... sure, it's all there in the vector maths, but I personally couldn't make head or tail of it until I could imagine some physical models to attach it all to - good luck
P.S. What happens if they're *not* perfectly conductive... like copper...? The analysis rapidly gets messy in conductive media and all sorts of interesting effects arise such as the electric field component can now be non-zero at the plate boundary (tiny, sure - but not zero, owing to their finite resistance). This allows an (exponentially decaying) magnitude to exist throughout the metal thickness which gives rise to coupling through the plates etc etc. Lots of fascinating applications! (Underwater comms, for example).