How can I take advantage of meeting myself?

I don't know that we'll ever know, seeing as the secret has been part of the game for 15 years and nobody's quite figure it out yet. Having said that, my best guess is that it has something to do with this:

What does it mean when I have done nothing to effect time and all of the sudden a temporal storm happens.?

This is an indication that an alternative version of yourself, one that has been sent in to assist the microids, has done something to effect the timeline of one of your friendly species. He is your alter-ego.

What I take from this is that events in another game you've played can effect the game you're currently playing. In this case, it's likely helpful if you can remember what you did in that game, so you can predict the effects in your current game. I'd imagine this transmission you can make to yourself is intended for this purpose. You could use it to determine which other game is impacting your current game.

Since it's not always possible to achieve success in the game, due to the fact that the two goals are frequently divergent, having two chances in the same universe might make it possible to do both, but this is mostly just speculation on my part.

When meeting yourself for the first time, your alternate self will ask you who you are. Respond by saying I am you. After this you will be able to enter a number.

When encountering yourself again ask yourself who you are, and your other self will respond with "I am you". Then enter the numbers you originally communicated to yourself.

Go to the teleportation screen, it's now possible to receive a special weapon trough transport from yourself; the Microid diffusion weapon.

Source: series of youtube movies, in particular this one.