You should read the document "Dominus Iesus" - it is easily found online. It discusses the theology relevant to this discussion.
To put the whole thing in a nutshell: The Second Person of the Trinity - the Son, the Logos - became incarnate in order to reveal God fully to mankind; Jesus in the course of this time on earth established one Church, to which He promised to give His Spirit to guide and His protection against failure; Because there is only one Logos Who became incarnate and one Spirit, they are working solely through this one Church. Only the Catholic Church maintains all of the marks of this Church founded by Christ (i.e. "the Church of Christ subsists in the Catholic Church" as Lumen Gentium 8 explains).
This being the case, other Christian communities are not the Church Christ founded - though they share elements of the Catholic Church and are relevant to the salvation of the Baptized Christians in them. Communities that have maintained a valid Episcopate and valid Eucharist are properly called particular Churches. However, the efficacy of all of these groups is dependent on the grace entrusted to the Catholic Church.
Non-Christian religions also have 'seeds of truth' in them that prepare their adherents for the Gospel, but by virtue of their other beliefs and practices hinder (but do not prevent) the salvation of their adherents. Thus it is beneficial (and indeed necessary) for the Church to evangelize.
No, the idea of "Extra Ecclesiam Nulla Salus" does not equate to "only card-carrying Catholics can be saved" - but all salvation still comes through the Church, and only through the Church.
“Only the Catholic Church maintains all of the marks of this Church founded by Christ”
This is why I converted, when I realised the argument wasn’t, “X is acceptable”, but whether X or Y has the fullness of Christ.
What I meant is that the Church doesn’t teach that you must be a member of the Catholic church in order to be saved.