some intuition
Soft doubling is a tough one to understand. Take A7 vs. 6. If you get a ten, it doesn't affect your hand. Getting A,2, or 3 improves your hand, but getting 4,5,6,7,8, or 9 makes your hand worse! Therefore it seems that you shouldn't double: twice as many cards make your hand worse as make it better. And in fact, that's the truth (as Parker said): if you take a hit, you will, on average, make your hand worse. So why should you double?
In exchange for making your hand worse, you are given the chance to put more money on the table, so even if you make your hand worse, that's not necessarily a bad thing. Would you rather have $10 riding on a hand that has an edge of 28%, or have $20 riding on a hand with an edge of 19%? The first hand is worth $2.80, while the second is worth $3.80! That sort of thing occurs with soft doubles. (In fact, those percentages are the exact numbers under consideration for A7 vs. 6 in a six-deck, S17 game!)
Whenever you double-down, you are giving up the flexibility of making your hand as good as possible, but in exchange, you are allowed to put more money down. If it's basic strategy to double down, it's because the one effect outweighs the other. That's it.
Lefty