They sign these "minor league" contracts because they are low risk. Many times, they will include an "invitation to spring training". By signing a player in this manner, they dont have to clear a roster spot for them (thereby taking a chance at losing a decent prospect) and should the player not make the cut, he can be sent to the minors without any waiver trouble (though, some veterans choose not to report).
This is typically used on fringe major leaguers , veterans in the "downside" of their career or by franchises looking to audition a variety of players at a particular position (example, 5 players on minor league deals give them more flexibility to try each at a position, rather than 2 or 3 on a major league roster). If a player on this deal is called up to the big leagues, they make the roster move when the call up is made, so they dont make a mistake in signing a player who cant play at the level they want.
In Clemens case, the Yankees wanted time for him to get in shape, and this provides it as they dont need to add him to the roster until they want him to join the team. In Belliards case, they wanted to audition a variety of players to replace Vidro.