Science is not about absolute proof, but about evidence. The goal of an experiment is to provide evidence that can then be verified by other researchers to add to the knowledge within the field. The experiment described provides evidence that the subject projected her consciousness outside of her physical body, but it's not clear how strong that evidence is.
First, given the structure of the experiment, there were multiple chances for the participant to inadvertently be exposed to the "random" number (even though the chances of that happening are very small). The design would have been stronger if the "random" number were selected by a third party previous to the experiment and if it were placed in a location where it could not be read by either the participant or the researcher. This is called a "double-blind" experiment.
Second, it would be pretty simple to set up a video camera in a room to monitor the participant's movements during the night. This would help to eliminate fears of cheating or peaking.
Third, it's not clear why he didn't try to repeat this experiment to increase it's validity.
Having said all of this, the chances of correctly guessing a 5 digit random number with 4 chances (four nights of sleeping) are 1 in 25,000. Those are pretty slim odds, and, if the proper controls were in place, this would provide significant evidence that the participant projected her consciousness outside of her physical body in order to read the number on the shelf.
Contrary to what many scientists will tell you, results that occur in isolation and have not been repeated can provide evidence. Repeatability provides stronger evidence and adds reliability and validity to the experiment. Without repeatability, the results cannot be dismissed or discarded, but it must be considered that the results could be an anomally that may not reflect a trend. Many people in different science (especially the social sciences) now turn to meta-analysis where the results of many experiments are combined to help get a larger sample size and to help demonstrate that results have been repeated. At best, this experiment can be considered to be one experiment that would have to be included in a meta-analysis in order to produce evidence that is likely to be accepted as scientific evidence.
This type of work is typically now referred to as Remote Viewing, even though this experiment is being done within a single room. I've included some information below concerning remote viewing as a reference.