Like everything else in life, designing engines involves compromises.
If the gaps between the turbine blade tips and their casings are too large, a lot of gas will leak around them instead of passing through the turbines. This will reduce the efficiency of the turbines. So efficiency is best when these gaps are small.
But turbine blade length does not remain constant. When running with high rpm and high temperatures such as during take-off, thermal expansion causes the blades to expand. So if the gaps are too small when the engine is built, the blades are likely to rub against the casing when running hot and fast.
The combined effects of high temperatures and high rpm also cause blade creep, which results in permanent increases in their length. The amount of creep increases with hours run, so the blades gradually become longer as engine age increases. If the gaps are too small when the engine is built, creep will cause the blades to rub against the casings. So for a long engine life without turbine tip rub, the gaps at the tips need to be reasonably large when the engine is built.
Active clearance control provides a useful compromise by shrinking the casings by the correct amount to match engine running conditions. This is achieved by blowing cool air onto the outside of the casings. This cools the casings, causing them to contract, thereby reducing tip clearances.
During take-off the blades are comparatively long, so very little contraction of the casing is required. So fairly hot air (from the 9th stage of the compressor for example) is used for the coolling process.
In the cruise the lower rpm and turbine temperatures mean that the blades are shorter and the gaps larger. So in cruise flight a greater degree of contraction is achieved by blowing cooler air (from the 4th stage of the compressor for example) over them.
Different engine types may use different sources of cooling air, but the general principle is the same.