There are four aspects (termed conditions within the article) of a patient's history that influence the likelihood of their returning to hospital after a previous stay there. Each aspect is scored according to suitable criteria. To take just one example, if the patient has been admitted to an emergency department once/twice/three times in the previous six months, the score for this condition is respectively 1/2/3.
The term "cross-condition" is adjectival, merely describing the LACE index as a combination of the scores for all four conditions (i.e. a score that crosses all conditions). The nature of the index is that it does not depend directly on any single condition but on a simple arithmetic combination (in this case, the sum) of scores for each of the conditions. A cross condition index is therefore nothing more than an index (score) composed from the scores of each of the four contributing conditions.
I add that this technique of reducing scores from a number of variables to a single score somehow hoped to represent the whole set is common when presenting complex data in a simple way. Key Performance Indicators in business may combine measures of several things (fraction of faulty goods, number of complaints, average invoice payment time ...). Environmental Quality Indices combine several measures of environmental health in one scale that is derived (not necessarily by simple totaling of individual scores) from them in various ways. I do not quote references because you may easily find example on a search for "Quality index". These indicators are all equivalent to cross-condition indexing as in the article you quote.
In simplest summary, the cross-condition index (or score) is the total of the scores across all the four conditions. For example, scores of [2,5,3,2] give a cross condition score of 12.