Well I guess the main question would be whether you are trying to convert an existing application or building a new one. If you are building a new application or just looking a conceptually background than that is simple enough to explain.
A web service is nothing more than a http request that send back some type response. In some cases that response could be XML others JSON or really any data format. The only thing you gain from using a well known format like XML or JSON is that most application languages provide a means to easily parse and extract the data through various APIs of thier own. Either included internally or provided by some type of additionally library. The reason you would do do something like this though is to completely separate the GUI of an application from the mechanism used to fetch and persist data to what ever end data source you are using. In some cases you might just even use a vendors web services API. When it comes to vendors using web services provides a means of exposing data management mechanisms to the outside world without providing users direct access to data facility itself or hardware which it is hosted which would be a HUGE security vulnerability. Instead an API is created to expose the data separate from the data store and/or hardware to management it.
I myself am kinda stumped as to what the quote "I was told from a lecturer that web services could be used in place of XSLT to create a web application" actual means. I mean that is a pretty generic thing to say. The word "web services" could really replaced with any technology such as; ASP.NET, PHP, Ruby, JSON, etc and still make sense. Could you perhaps provide a little more details as to the context of the quote itself.