There is a related (and more important) reason than arbitrary precision: Geohashes with a common prefix are close to one another. The longer the common prefix, the closer they are.
54.321 -2.345 has geohash gcwm48u6
54.322 -2.346 has geohash gcwm4958
(See http://geohash.org to try this)
This feature enables fast lookup of nearby points (though there are some complications), and only works because we interleave the two dimensions to get a sort of approximate 2D proximity metric.
As the wikipedia entry goes on to explain:
When used in a database, the structure of geohashed data has two
advantages. First, data indexed by geohash will have all points for a
given rectangular area in contiguous slices (the number of slices
depends on the precision required and the presence of geohash "fault
lines"). This is especially useful in database systems where queries
on a single index are much easier or faster than multiple-index
queries. Second, this index structure can be used for a
quick-and-dirty proximity search - the closest points are often among
the closest geohashes.
Note that the converse is not always true - if two points happen to lie on either side of a subdivision (e.g. either side of the equator) then they may be extremely close but have no common prefix. Hence the complications I mentioned earlier.