For the first part of your question about
Eager learning methods use all available training examples to build a classiﬁer
in advance that is later used for classiﬁcation of all query instances.
Lazy learning, or instance based learning, is a learning method that delays the building of the classiﬁer until a query is made to the system. The algorithm is trying to find similar examples from the training data to the query and uses them to build hypothesis for the classification. Examples that are used are localized near the query by some similarity.
For example if we have points in the plane that are classified with (+) sign and (-) sign, the eager learning will build single rule about how to classify any new point. In contrast, the lazy learning will aproximate only the nearest points signs to predict what will be the sign of the new point.
For the second part about
I guess that here the author means the online learning. This means that each new query is added to the traing data after its value is known.
Because of this, the eager learning must update its hypothesis after each new query and should process each query one at a time.
In contrast, the lazy learning can take many simultanious queries (if they are not locally close) because it uses only the examples, locally close to it.