First of all, as the other answers have indicated, Scalding Tarn does not produce any mana, it just lets you search for another land that can. The colors on this card have absolutely nothing to do with its ability. The blue and red are just there as a visual reminder that the ability lets you search for an Island or a Mountain. The card could be orange, or pink, but the ability would still be the same.
So, down to the real question: why is its activated ability useful, especially since you have to pay a life?
Here are a few benefits/uses:
That Scalding Tarn in your hand could turn into, for example, Volcanic Island, Plateau, Underground Sea, or just a basic Mountain or Island. You have the flexibility to search for whatever land you may need to cast stuff.
Let's say in your opening hand you have a Scalding Tarn, a Lightning Bolt, and a Spell Snare. Play the Scalding Tarn, then you can wait to "crack" the fetchland until the last second. If your opponent plays a creature, grab a Mountain to cast Lightning Bolt; if they play a 2cmc spell, grab an Island to cast Spell Snare instead. Flexibility!
Shuffling: Let's say you play a Brainstorm, and don't really like the cards you put back on top (maybe they're all lands that you don't need). Crack Scalding Tarn to shuffle your library! Better to shuffle hoping that you'll get something worth drawing than to sit there for a few turns knowing you'll draw crap that doesn't help you win.
Landfall: Let's say you have a Steppe Lynx in play. Then play your Scalding Tarn, immediately crack it, and grab some land you need. Two lands entered play, so now Steppe Lynx is a 4/5 monster (for this turn) that only cost you 1 mana and 1 life. Totally worth it. Landfall + Fetchlands can give you some bonkers plays.
Graveyard: By cracking the fetch, you put a card in the graveyard, which can benefit things like Grim Lavamancer/Tarmogoyf/Deathrite Shaman
Every time you search your library for a land, you reduce the chance of drawing a land the next time you draw. This is nice later in the game, when lands are typically dead draws.
Statistically, the "thinning" effect of a single fetchland doesn't make a big difference, but multiple copies can. If you play fetchlands and your opponent doesn't, you'll have a slight edge in terms of drawing potential threats (instead of lands), but at the cost of some of your life.
Remember, cards like Necropotence taught us long ago that life is just another resource; using your resources wisely will win you games. That being said, sometimes the life loss associated with fetchlands may not be wise. Some tournament-worthy decks run 6-8 fetchlands, others run 1-2, some run none at all. It depends entirely upon your deck versus the metagame you've chosen to play in.
IMHO, fetchlands are a much better choice when used for flexibility and/or synergy than for just deck thinning alone, but that's just me. You get the deck thinning automatically, but the flexibility/synergy comes with intelligent deck construction.
Anyway, if you don't want those Scalding Tarns, I'll take them. Aside from being good cards for all of the reasons I've just described, those suckers are worth something like $70 each right now!