James Clerk Maxwell's equations on electromagnetism seem to rely on the existence of the aether and he was the one which suggested the use of an interferometer to measure the speed of a frame of reference with respect to this 'ultimate rest frame' -- later tested by Michelson and Morley and the 'null result' obtained.
How could this be explained?
1. Maxwell was right, but Galilean relativity was wrong
2. Maxwell was wrong, Galilean relativity was right
3. Both were right and something else happened (eg Lorentz proposed that the moving objects are contracted in direction of motion) -- this is what was thought by the physicists of the day
The Lorentz transformation eqns (basically the 'time dilation', 'length contraction' etc...) formulae were used to describe the changed in the object
Einstein figured out that these eqns could be applied to space and time itself, rather than to the object and the application of these transformation eqns to space and time could explain the 'null result' obtained by MM.
Einstein thought, although Maxwells equations seemed to rely on the existence of this 'preferred reference frame' (aether), why would electromagnetic interactions need this to be defined, but dynamic interactions (i.e., gravity, newtons laws) do not? Einstein no likey this mathematical contradiction of these different interactions. Einstein showed that the Lorentz transformations were enough to explain length contraction etc... provided that Galilean Relativity (i.e., Velocity of A relative to B = Velocity of A - velocity B) was not entirely accurate.
So basically, What Lorentz and others thought (before Einstein published SR) was the Lorentz transformations were applied to the objects, in the presence of the aether
Einstein said, the aether is not needed to explain the transformations, if the transformations actually occur within spacetime (something Lorentz and Poincare rejected), therefore rendering the aether unneeded to explain the occurences, hence rendering it superfluous.
On a side note, in 1920 when Einstein came up with General relativity, he re-introduced some of the aspects of this magical 'aether'