Question: OTM (out-of-the-money) options are cheap but even decent move can yield dramatic returns that eclipses ATM (at-the-money) or ITM (in-the-money) options, but what I don't understand is that delta and gamma is supposed to be low for OTM so what causes the giant leap in returns when price approaches the strike? Often, it doesn't even need to touch the strike but a large swing can drive up the premium.

Near expiry options (1 week let's say) are also priced cheaper, but does it contribute to the high leverage?

Answer: 
I have a feeling you're looking at large *percentage* increases which is very different that *absolute* increases. Delta and Gamma measure *absolute* changes in value per dollar of change in the underlying, not *percentage* changes. If a option that costs 5 cents goes up another 5 cents, that's a 100% increase, but a $1 option that goes up 5 cents is only a 5% increase. If the 5 cent option goes *down* 2 cents, though, it's a 40% *drop*.

So yes, you can see huge swings in relative returns just because the prices are so small relative to ITM options, but those swings go both ways.

The other problem is that the *reason* those options are so cheap is because there's very little chance that they'll be worth anything at expiration.

Note that this has nothing to do with *leverage*, where returns are multiplied because you borrow money to multiply the amount of money invested, it's just the mathematical phenomenon of larger percentage changes when the denominator is smaller.
AFAIC, understanding the statistical underpinnings of Black Scholes etc. and the Greeks (other than delta) isn't relevant for most retail option traders. It's like asking someone for the time and they tell you how to build a watch :->). In this case, all you need to grasp is that options near the money benefit the most percentage wise from price change.

Let's use Joe Taxpayer's option prices for stock XYZ. Suppose one second after his pricing, XYZ jumps $2.50 with no change in implied volatility. Take the option prices and shift them down one row. I priced the missing ones. Column 1 is the strike price. Column 2 is his original call premiums. Column 3 is the new option price after a $2.50 rise in XYZ. Column 4 is the call's gain in dollars. Column 5 is the call's gain in percent. I hope this uploads properly.

XYZ = $178

175.00 2.15 3.70 $1.55 72%

177.50 0.95 2.15 $1.20 126%

180.00 0.35 0.95 $0.60 171%

182.50 0.14 0.35 $0.21 150%

185.00 0.07 0.14 $0.07 100%

187.50 0.04 0.07 $0.03 75%

190.00 0.03 0.04 $0.01 33%

As you can see, the ITM options have the larger dollar gain and the near the money have the larger percent gain.  

 .

Suppose XYZ jumps another $2.50 (total of $5). Again, the same holds true.

175.00 2.15 5.75 $3.60 167%

177.50 0.95 3.70 $2.75 289%

180.00 0.35 2.15 $1.80 514%

182.50 0.14 0.95 $0.81 579%

185.00 0.07 0.35 $0.28 400%

187.50 0.04 0.14 $0.10 250%

190.00 0.03 0.07 $0.04 133%

Options provide leverage because you control 100 shares with a small premium, compared to the cost of the underlying. And if you compare an equi-dollar investment, it offers multiples of that (compare the returns from buying one $175 call with buying six $180 calls).

And yes, near expiry options contribute to the leverage because they have less time premium to lose as time premium decreases and intrinsic value increases as share price increases.