Any election method that complies with the Smith criterion also complies with the Condorcet winner criterion, since if there is a Condorcet winner, then that winner is the only member of the Smith set. Voting methods that pass the Smith criterion are called "Smith-efficient" (or often "Smith-efficient Condorcet methods"). They also pass the mutual majority criterion, as the Smith set will always be a subset of the mutual majority-preferred set of candidates (each of the mutual majority-preferred candidates pairwise beats all non-mutual majority-preferred candidates by a majority, but some of them may pairwise beat each other, so the Smith set will at most be all of them.)
Methods that do not comply with the Condorcet winner criterion, such as Approval voting, Cardinal Ratings, Borda count, Plurality voting, and Instant-Runoff Voting, also do not comply with the Smith criterion.
Any voting method can be made Smith-efficient by first eliminating everyone outside the Smith set and then running the voting method. Another Smith-efficient approach may be to eliminate everyone outside the Smith set, then among the remaining candidates, eliminate the candidate the voting method considers worst, and repeat. See Tideman's Alternative methods#Notes for more information.
Schulze has proposed a multi-winner generalization of the Smith criterion which can be roughly described as: "if for a group of candidates equal to or smaller than the number of winners, a certain number of them would always win when up to that certain number of any of them and enough of the candidates not in the group face off such that there are always at most (one more than the number of winners) candidates running, and the same holds for any number smaller than the certain number, then at least that certain number of candidates from the group must be in every winner set in the Smith set, and the voting method must pick one of the Smith winner sets as the final winner set." 
Note that this should be considered as only one of several conditions a voting methods should pass to be considered a Smith-efficient Condorcet PR method, since Bloc Score Voting passes it yet is not proportional or a Condorcet method.
One way to argue for the Smith criterion is that not electing from the Smith set means a plurality of voters, who would presumably have the power to force their preferred candidate in the Smith set to beat whoever actually won in many voting methods, do not get their preference. In essence, it's a criterion that makes the most sense for voters who want maximal power. However, note that the Smith set itself can be made more utilitarian if voters tend to express weak preferences as equal rankings, or are allowed to express weak preferences in the head-to-head matchups. See Ballot#Notes and Asset voting for discussion on this.
The Smith criterion can alternatively be worded as "any time the candidates can be split into two groups such that every candidate in the first group pairwise beats every candidate in the second group, then one of the candidates in the first group must win."