The units here are relative units of intensity. There may be about a picomole of probes on a microarray spot, but the units of intensity are not scaled to the precise concentration of DNA on the spot.
There are many variables which make exact measurements of intensity difficult to translate into the number of RNA bound to a spot. The main one is nonspecific or non-target binding. The oligomer on the array surface is typically between 25 and 50 bases long - it isn't the entire length of the RNA being tested by a given spot. Another RNA may have some or most of the same sequence as a probe, or some labelled RNA will tend to get associated with some spots. Usually factors such as the position of the spot on the array surface, the GC composition of the probe and variations in the batch of RNA preparation will also cause the intensity to vary.
Still, even with all this, with the same prep conditions and with an identical set of arrays, its valid to look at the intensity over time. In this paper, the investigators took array measurements at different time points and at two different temperatures after 4-Thiouracil is added to the cells, which allows one to follow RNA produced by transcription at a given time.
Since the units are arbitrary but still proportional to the mRNA found in the cells, a difference experiment like this is valid, even if the absolute concentration of any given RNA is still a bit of a mystery.