What is the real meaning of length contraction?

No it really is a length contraction - but this is easier to see with the classic example of the measured cosmic ray muon flux.

High energy muons shower down on Earth from the upper atmosphere:

• Muons have a mean lifetime of approximately 2.2 microseconds in the laboratory.
• The distance muons needs to travel from the upper atmosphere to the Earth is approximately 15km.

Let's consider two situations:

1. Before taking length contraction into account:

A muon travelling close to the speed of light would be expected to travel approximately 660m before decaying. Hence we wouldn't expect the measure any muons at ground level. However the measured flux of muons at ground level is actually 1 cm$^{-2}$ min$^{-1}$... so where are these muons coming from?

2. After taking length contraction into account:

If you consider a muon with energy 20 GeV, it has a length contraction factor of 189 - so the distance that a 20 GeV muon observes is from the atmosphere to the earth not 15km, it is 79m! The length has contracted. Hence you would expect the majority of muons at this energy to survive - which is what is observed.