Time dilation and length contraction are not illusions in the theory of special relativity (SR), but rather real physical effects that arise from the fundamental principles of the theory. While they can seem counterintuitive, they have been extensively tested and observed in various experiments, including the muon experiment you mentioned.
In the muon experiment, high-energy cosmic ray muons, which are short-lived particles, are created in the upper atmosphere and travel towards the Earth's surface. According to an observer on the ground, the muons should decay before reaching the surface due to their short lifetime. However, many muons do reach the surface, which can be explained by time dilation and length contraction.
From the perspective of an observer at rest on the Earth, the muons appear to experience time dilation. This means that the muons' internal clocks appear to run slower compared to the clocks of the stationary observer. As a result, the muons' lifetimes are effectively extended, allowing them to travel a longer distance before decaying.
Simultaneously, from the perspective of the muons, the length of the distance they need to travel (the atmospheric height) appears contracted due to length contraction. This contraction makes the distance shorter in the muons' frame of reference, allowing them to cover the distance more quickly.
The key point here is that time dilation and length contraction are relative effects that depend on the relative motion between observers. Each observer experiences their own proper time and measures lengths according to their own frame of reference. The observed time dilation and length contraction are not illusions but real consequences of the relative motion between frames of reference.
It is important to note that while these relativistic effects may seem paradoxical at first, they have been consistently confirmed by numerous experiments and have practical implications in various areas of modern physics, including particle accelerators, GPS systems, and high-speed particle interactions. The theory of special relativity has proven to be an extremely accurate and reliable framework for describing the behavior of objects moving at significant fractions of the speed of light.