In basic open leagues, you can only play the upcoming series live and there are not a lot of other people in those leagues who play live. You may be able to play some of those games live if you contact owners ahead of time, especially if you notice which owners occasionally play live by looking at your league standings page and seeing whether anyone shows up as "ON." You can post a message on the league standings page asking whether anyone would play some live games when they come up on the schedule.
You can usually find live players more than willing to play some exhibition games with you so you can try out the live feature. Log into the live program and just post a message that you want to give live a try, and someone will likely be willing to play an exhibition game with you. The first time you play a live game, it can be a little frustrating figuring out how to make or accept a challenge, but anyone who is online will be able to help out. You can also try it out yourself by going into one of the "Open Play" rooms and playing against the computer. (The SimLeague Venue is where scheduled league games are played; Open Play is for exhibition games, in which you can use a team you created or any team from MLB history).
If there's a specific time you know you'll be on and looking to try an exhibition game, post that here and chances are someone will be able to get on and walk you through the steps to play.